Tropical plactic algebra, the cloaktic monoid, and semigroup representations

TL;DR

This paper introduces a tropical plactic algebra and related monoids, providing new linear representations via tropical matrices, and develops combinatorial tableaux tools to analyze semigroup identities and algebraic properties.

Contribution

It defines a tropical plactic algebra and monoids, linking combinatorial tableaux with algebraic representations, and proves these monoids satisfy identities of tropical triangular matrices.

Findings

Faithful linear representations of monoids by tropical matrices.

Development of configuration tableaux for encoding Young tableaux.

Proof that monoids satisfy identities of tropical triangular matrices.

Abstract

A new tropical plactic algebra is introduced in which the Knuth relations are inferred from the underlying semiring arithmetics, encapsulating the ubiquitous plactic monoid $\mathcal{P}_n$. This algebra manifests a natural framework for accommodating representations of $\mathcal{P}_n$, or equivalently of Young tableaux, and its moderate coarsening -- the cloaktic monoid $\mathcal{K}_n$ and the co-cloaktic $ ^{\operatorname{co}}\mathcal{K}_n$. The faithful linear representations of $\mathcal{K}_n$ and $\, ^{\operatorname{co}} \mathcal{K}_n$ by tropical matrices, which constitute a tropical plactic algebra, are shown to provide linear representations of the plactic monoid. To this end the paper develops a special type of configuration tableaux, corresponding bijectively to semi-standard Young tableaux. These special tableaux allow a systematic encoding of combinatorial properties in numerical algebraic ways, including algorithmic benefits. The interplay between these algebraic-combinatorial structures establishes a profound machinery for exploring semigroup attributes, in particular satisfying of semigroup identities. This machinery is utilized here to prove that $\mathcal{K}_n$ and $\, ^{\operatorname{co}} \mathcal{K}_n$ admit all the semigroup identities satisfied by $n \times n$ triangular tropical matrices, which holds also for $\mathcal{P}_3$.

Equations

$$\begin{array}[]{cccc}&a\;c\;b=c\;a\;b&\text{if}&a\leq b<c\;,\\[2.84526pt] &b\;a\;c=b\;c\;a&\text{if}&a<b\leq c\;.\\ \end{array}$$

$$\big{(}\mathfrak{a}+\mathfrak{b}\big{)}^{m}=\mathfrak{a}^{m}+\mathfrak{b}^{m}\qquad\text{for any }m\in\mathbb{N}.$$

$${\overrightarrow{\mathrm{len}}}_{{\mathcal{A}}_J}(u)={\overrightarrow{\mathrm{len}}}_{{\mathcal{A}}_J}(v)\text{for every convex sub-alphabet AJ of AI,}$$

$$ð:{\mathcal{P}}_I\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\twoheadrightarrow {\mathcal{K}}_I$$

$${{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(a_{\ell }):=a_n{a}_{n-1}\cdots a_{n-\ell +2}{a}_{n-\ell }\cdots a_1,a_{\ell }\in {\mathcal{A}}_n,$$

$${{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(w):={{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(a_{{\ell }_1})\cdots {{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(a_{{\ell }_m}),w=a_{{\ell }_1}\cdots a_{{\ell }_m},$$

$${\partial }_1:\text{lhook}{\text{joinrel}}^{\ast }{\text{joinrel}}_n\text{joinrel}\mathcal{A}\text{joinrel}\text{relbar}\text{relbar}\text{relbar}\text{relbar}\to {({\mathcal{A}}_n^{\ast })}^{[1]}\subset {\mathcal{A}}_n^{\ast },w⟼{{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(w).$$

$${\partial }_1\text{lhook}{\text{joinrel}}_n\text{joinrel}\mathcal{P}\text{joinrel}:\text{joinrel}\text{relbar}\text{relbar}\text{relbar}\text{relbar}\to {({\mathcal{P}}_n)}^{[1]}\subset {\mathcal{P}}_n$$

$$u{\equiv }_{\mathrm{clk}}^{\mathrm{co}}v\iff {\partial }_1(u){\equiv }_{\mathrm{clk}}{\partial }_1(v).$$

$$\mho :{\mathcal{K}}_n\stackrel{\sim }{\text{relbar}\text{relbar}\text{relbar}\text{relbar}\to }{\mathfrak{A}}_n^{\times }.$$

$${\Pi }_{(C,p,q)}:\tilde{w}x\tilde{w}=\tilde{w}y\tilde{w},$$

$${\wp }_n:{\mathcal{P}}_n\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\twoheadrightarrow {\mathfrak{A}}_n^{\times }\times {}^{\diamond }\text{joinrel}{\mathfrak{A}}_n^{\times }$$

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$$a\le b\text{implies}ca\le cb,ac\le bc$$

$$w=a_{{\ell }_1}^{q_1}\cdots a_{{\ell }_m}^{q_m}\in {\mathcal{A}}_I^{\ast },{\ell }_t\in I,q_t\in \mathbb{N},$$

$$\varphi (a\cdot b)=\varphi (a)\cdot \varphi (b)$$

$$\varphi :\mathcal{M}\text{relbar}\text{relbar}\text{relbar}\text{relbar}\to {\mathcal{A}}_I^{\ast }{/}_{\Xi },\Xi \subset {\mathcal{A}}_I^{\ast }\times {\mathcal{A}}_I^{\ast }.$$

$$\mathrm{rev}:{\mathcal{A}}_I^{\ast }\text{relbar}\text{relbar}\text{relbar}\text{relbar}\to {\mathcal{A}}_I^{\ast },\overrightarrow{w}⟼\overleftarrow{w},$$

$${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}_{n}(a_{\ell}):=a_{n}a_{n-1}\cdots a_{n-\ell+2}a_{n-\ell}\cdots a_{1}=\prod^{1}_{\scriptsize{\begin{array}[]{c}t=n,\\ t\neq n-\ell+1\end{array}}}\hskip-11.38109pta_{t}.$$

$${{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(w):={{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(a_{{\ell }_1})\cdots {{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(a_{{\ell }_m}),$$

$${{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(w):={{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(u){{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}}_n(v),$$

$$\mathrm{cmr}:{\mathcal{A}}_n^{\ast }\text{relbar}\text{relbar}\text{relbar}\text{relbar}\to {\mathcal{A}}_n^{\ast },w⟼{}^{\mathrm{co}}\text{joinrel}\text{joinrel}\mathfrak{M}(w),$$

$${\mathcal{A}}_n^{\ast }\supset {({\mathcal{A}}_n^{\ast })}^{[1]}\supset {({\mathcal{A}}_n^{\ast })}^{[2]}\supset {({\mathcal{A}}_n^{\ast })}^{[3]}\supset \cdots ,$$

$${\mathcal{A}}_n^{\ast }\stackrel{{\partial }_1}{\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\twoheadrightarrow }{({\mathcal{A}}_n^{\ast })}^{[1]}\stackrel{{\partial }_2}{\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\twoheadrightarrow }{({\mathcal{A}}_n^{\ast })}^{[2]}\stackrel{{\partial }_3}{\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\twoheadrightarrow }{({\mathcal{A}}_n^{\ast })}^{[3]}\stackrel{{\partial }_4}{\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\text{relbar}\text{joinrel}\twoheadrightarrow }\cdots .$$

$$\Pi :u=v,$$

$$\varphi (u)=\varphi (v)$$

$${\Pi }_{(C,p,n)}:{\tilde{w}}_{(C,p,n)}\underline{x}{\tilde{w}}_{(C,p,n)}={\tilde{w}}_{(C,p,n)}\underline{y}{\tilde{w}}_{(C,p,n)}.$$

$$x:=\widecheck{x}\ {\widecheck{y}}\$$

$${\check{\Pi }}_{(\check{C},p,n)}:{\check{w}}_{(\check{C},p,n)}\check{x}\check{y}{\check{w}}_{(\check{C},p,n)}={\check{w}}_{(\check{C},p,n)}\check{y}\check{x}{\check{w}}_{(\check{C},p,n)},$$

$${\Pi }_{(C,2,2)}:y{x}^2{y}^{2}x\underline{x}y{x}^2{y}^{2}x=y{x}^2{y}^{2}x\underline{y}y{x}^2{y}^{2}x.$$

Full text

The research of the author has been sported by the Research Councils UK (EPSRC), grant no EP/N02995X/1.

Institute of Mathematics, University of Aberdeen, AB24 3UE, Aberdeen, UK.

 Email: zzur@abdn.ac.uk.

1. 1 Preliminaries
2. 2 Tropical plactic algebra
3. 3 Tropical matrices
4. 4 Troplactic matrix algebra
5. 5 The cloaktic  and the co-cloaktic monoids
6. 6 Configuration tableaux
7. 7 Representations of tableaux and of the plactic monoid
8. 8 Remarks and open problems
9. A Proofs and completions to §7.1

Introduction

This paper introduces linear representations of the plactic monoid and its coarsening, called the cloaktic and the co-cloaktic monoids, together with a new encapsulating algebra. The plactic monoid $\mathcal{P}_{I}=\mathsf{PLC}(\mathcal{A}_{I})$ is the presented monoid $\mathcal{A}_{I}^{*}/_{\equiv_{\operatorname{knu}}}$. That is, the quotient of the free monoid  $\mathcal{A}_{I}^{*}$ over an ordered alphabet $\mathcal{A}_{I}$ by the congruence $\equiv_{\operatorname{knu}}$ determined by the Knuth relations (also called plactic relations)

[TABLE]

This monoid was first appeared in the context of Young tableaux (Knuth [26] and Schensted [45]) and has been followed by an extensive study of Lascoux and Schützenberger [30], establishing an important link between combinatorics and algebra via its bijective correspondence to semi-standard Young tableaux  [25].

The combinatorics of $\mathcal{P}_{I}$ is framed by Young tableaux and has been studied mainly in traditional perspective (e.g., Gröbner-Shirshov bases  [4]). It has various applications (e.g., in symmetric functions  [37], Kostka-Foulkes polynomials [31], and Schubert polynomials [32]). Related representations have looked at monoid algebras $K[\mathcal{P}_{I}]$ over a field $K$ for a finite $I$ [7, 27]. At the same time, Young tableaux has been found to be an important characteristic patterns in classical representation theory of groups [8, 10], especially in representations of the symmetric group  [44] and algebraic combinatorics  [36], as well as in combinatorics   [35] and computer science [41]. Therefore, in this challenging arena, a direct algebraic description of the plactic monoid strengthens the mutual connections to classical representation theory, providing an additional algebraic-combinatorial approach to group representations.

To address this goal, this paper introduces

• •

a semiring structure in which the Knuth relations (KNT) are inferred from its arithmetics,

• •

meaningful coarser monoids of the plactic monoid, and

• •

linear representations of the plactic monoid and its coarsening.

These objects pave a new way to study the plactic monoid and Young tableaux systematically. To develop linear representation, we are assisted by several auxiliary objects (i.e., semigroups, tropical matrices, digraphs, and configuration tableaux), enhancing the interplay among them.

The fundamental structure of the monoids of our main interest, determined as a quotient by multiplicative congruences, makes their analysis rather difficult. To approach such monoids straightforwardly, we adopt the familiar concept of associating an algebra to a multiplicative group which allows one to study groups in terms of algebras, e.g., the correspondence between Lie groups and Lie algebras. This paper applies a similar concept to the plactic monoid $\mathcal{P}_{I}$ by encapsulating $\mathcal{P}_{I}$ in the tropical plactic algebra $\mathfrak{plc}_{I}$ — a new semiring structure — abbreviated troplactic algebra. The underlying structure of this algebra is a noncommutative idempotent semiring (Defunition  2.4), generated by a set of ordered elements. In this algebra the Knuth relations (KNT) are intrinsically followed from the semiring arithmetics (Theorem   2.8).

An important property of $\mathfrak{plc}_{I}$ is that every product of its elements is uniquely decomposable to a sum of non-nested terms, each is a nondecreasing subsequence (Corollary 2.12). Furthermore, the generators of $\mathfrak{plc}_{I}$ admit the Frobenius, property (Lemma  2.15):

[TABLE]

A dual version of the troplactic algebra, denoted $\mathfrak{plc}^{\wedge}_{I}$, exists (Theorem 2.20) and is perfectly compatible with the co-mirroring map  (CMR) below.

An immediate consequence of the equivalence $u\equiv_{\operatorname{knu}}v$ of two words in $\mathcal{P}_{I}$ is that the lengths $\overrightarrow{\operatorname{len}}_{\mathcal{A}_{I}}(u)$ and $\overrightarrow{\operatorname{len}}_{\mathcal{A}_{I}}(v)$ of longest nondecreasing subsequences (subwords) in $u$ and $v$ are the same. Furthermore,

[TABLE]

which happens due to correspondence of $\mathcal{P}_{I}$ with semi-standard Young tableaux (Proposition 6.9). The converse implication, however, does not hold in general, which leads to defining the cloaktic monoid $\mathcal{K}_{I}=\mathsf{CLK}(\mathcal{A}_{I})$ as the presented monoid $\mathcal{A}_{I}^{*}/_{\equiv_{\operatorname{clk}}}$, a moderate coarsen of  $\mathcal{P}_{I}$. That is, $\mathcal{A}_{I}^{*}/_{\equiv_{\operatorname{clk}}}$ is the free monoid $\mathcal{A}_{I}^{*}$ subject to the congruence $\equiv_{\operatorname{clk}}$, determined by the relations (CLK). Then, the monoid homomorphism

[TABLE]

is surjective and preserves the congruence $\equiv_{\operatorname{knu}}$.

As expected, the homomorphism $\eth$ has a critical role in the construction of our linear representations of the plactic monoid, obtained from those of $\mathcal{K}_{I}$. Nevertheless, sometimes, $\mathcal{K}_{I}$ appeared to be too coarser, which leads us to pursuing additional representable monoids that respect the congruence  $\equiv_{\operatorname{knu}}$. To receive such a monoid, we restrict the framework to finitely generated monoids  $\mathcal{A}_{n}^{*}$ and introduce the co-mirroring of letters

[TABLE]

which induces the co-mirroring

[TABLE]

over whole $\mathcal{A}^{*}_{n}$. It also determines the (lexicographic) order preserving surjective homomorphism

[TABLE]

The homomorphism $\partial_{1}$ extends inductively to a chain of endomorphisms $\partial_{i}:{(\mathcal{A}_{n}^{*})}^{[i-1]}\lhook\joinrel\longrightarrow{(\mathcal{A}_{n}^{*})}^{[i]}$ of induced free monoids.

Our co-mirroring construction respects the congruence $\equiv_{\operatorname{knu}}$ (Theorem  7.2). Thus, for the finitely generated plactic monoid $\mathcal{P}_{n}$, the monoid homomorphism

[TABLE]

is injective — a monoid embedding. On the other hand, $\partial_{1}$ does not preserve the equivalence of $\equiv_{\operatorname{clk}}$ with the relations (CLK), but it determines the equivalence ${\equiv^{\operatorname{co}}_{\operatorname{clk}}}$, defined by

[TABLE]

The co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{CLK}}}(\mathcal{A}_{n})$ is constituted as ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}=\mathcal{A}_{n}^{*}/_{{\equiv^{\operatorname{co}}_{\operatorname{clk}}}}$, that is accompanied with the surjective monoid homomorphism ${{}^{\operatorname{co}}\joinrel\,\joinrel{\,\eth}}:\mathcal{P}_{n}\;{\relbar\joinrel\twoheadrightarrow}\;{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$, providing a second coarsening of $\mathcal{P}_{n}$.

The main goal of this paper is to construct linear representations of the plactic monoid $\mathcal{P}_{n}$ in terms of tropical representations of the cloaktic monoid $\mathcal{K}_{n}$ and the co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$. Therefore, a special focus is given to formulating the correspondences between $\mathcal{P}_{n}$ to $\mathcal{K}_{n}$ and ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ by relying upon Young tableaux.

We start by describing the combinatorial structure of the cloaktic monoid $\mathcal{K}_{n}$ in terms of a troplactic algebra  $\mathfrak{plc}_{I}$, obtained by using tropical matrices. Besides their conventional algebraic meanings, these matrices are also combinatorial entities, corresponding uniquely to weighted digraphs. As such, they intimately compose graph theory in tropical algebraic methodologies, exhibiting a useful interplay between algebra and combinatorics. The latter plays a major role in theoretical algebraic studies [2, 21] and in applications to combinatorics [3, 24], as well as in semigroup representations  [15, 19] and automata theory  [47, 48].

This mutual connection allows for producing a special class of tropical matrices that generate the finite tropical plactic algebra  $\mathfrak{A}_{n}$ (Theorem  4.2) along with recording lengths of the longest pathes in digraphs. In turn, these paths encode the longest nondecreasing sequences of the represented elements (Lemma  4.1). As being a troplactic algebra, the multiplicative submonoid $\mathfrak{A}^{\times}_{n}$ of $\mathfrak{A}_{n}$ immediately admits the Knuth relations  (KNT). More precisely, $\mathfrak{A}^{\times}_{n}$ is isomorphic to the cloaktic monoid  $\mathcal{K}_{n}$ (Theorem   5.6). Thus, it introduces the faithful tropical linear representation

[TABLE]

This isomorphism yields an efficient algorithm for computing the maximal lengths of all nondecreasing subwords of $w\in\mathcal{A}_{n}^{+}$ with respect to every convex sub-alphabet of $\mathcal{A}_{n}$ (Algorithm 5.14). Similarly, for the co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$, we obtain the monoid isomorphism \Omega:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}, whose image  ${{}^{\diamond}\joinrel{\mathfrak{A}_{n}}}$ is a dual troplactic matrix algebra $\mathfrak{plc}^{\wedge}_{I}$ (Theorem 5.19). Both isomorphisms $\mho$ and $\Omega$ are allocated with tropical characters that specify characteristic invariants.

An immediate result of the existence of the isomorphisms $\mho$ and $\Omega$ is that both monoids $\mathcal{K}_{n}$ and  ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ admit all the semigroup identities satisfied by the monoid ${\operatorname{TMat}}_{n}(\mathbb{T})$ of tropical triangular matrices (Corollaries  5.8 and 5.22). A particular form of these semigroup identities is constructed as

[TABLE]

where $\widetilde{w}:=\widetilde{w}_{(C,p,q)}$ is a fixed word over the variables $C=\{x,y\}$ that contains as factors all the possible words of length $q$ over $C$ in which no letter appears sequentially more than $p$ times, and such that the words $\widetilde{w}\,x\,\widetilde{w}$ and $\widetilde{w}\,y\,\widetilde{w}$ also satisfy this law [14, 15]. With this form in place, ${\operatorname{TMat}}_{n}(\mathbb{T})$ satisfies the identities (SID) with $p=q=n-1$ by letting $x=uv$ and $y=vu$. The identities (SID) generalize the Adjan’s identity of the bicyclic monoid [1], which is also faithfully represented by tropical matrices [19]. Also, ${\operatorname{TMat}}_{n}(\mathbb{T})$ satisfies a recursive version of Adjan’s identity [39].

With tropical representations of the monoids $\mathcal{K}_{n}$ and ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ in hand, our next goal is to formulate the surjective monoid homomorphisms $\eth:\mathcal{P}_{n}\;{\relbar\joinrel\twoheadrightarrow}\;\mathcal{K}_{n}$ and ${{}^{\operatorname{co}}\joinrel\,\joinrel{\,\eth}}:\mathcal{P}_{n}\;{\relbar\joinrel\twoheadrightarrow}\;{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ explicitly, and to explore their properties as reflected in the representations \mho:\mathcal{K}_{n}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}\mathfrak{A}^{\times}_{n} and \Omega:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}. To this end, we rely upon the well known one-to-one correspondence between the plactic monoid $\mathcal{P}_{n}$ and semi-standard Young tableaux $\mathsf{Tab}_{n}$ (e.g., see [29]). In a sense, tableaux are graphical patterns that accommodate symbols, i.e., letters of the associated plactic monoid, with a deep combinatorial meaning. Nevertheless, an additional machinery is required to canonically frame Young tableaux by tropical matrices and to convert visual-combinatorial information to suitable numerical-algebraic data.

This converting machinery is provided by $n$-configuration tableaux $\mathsf{CTab}_{n}$. That is, tableaux of fixed isosceles triangular shape that contain non-negative integers subject to certain structural laws, called configuration laws (Definition  6.11). Configuration tableaux correspond bijectively to semi-standard Young tableaux (Theorem   6.16) and are endowed with a self implementation of the Encoding Algorithm  6.19 that simulates the Bumping Algorithm  6.3 of $\mathsf{Tab}_{n}$. Their fixed shape enables the introduction of a canonical reference system, employed to state their correspondence to tropical matrices in $\mathfrak{A}^{\times}_{n}$ and ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$; thereby, to link tableaux with weighted digraphs. In this framework, encoding a letter $a_{\ell}\in\mathcal{A}_{n}$ in a configuration tableaux $\mathfrak{C}_{w}\in\mathsf{CTab}_{n}$ is interpreted as multiplying the matrix $\mho(w)$ by the matrix $\mho(a_{\ell})$ in $\mathfrak{A}^{\times}_{n}$ or, dually, $\Omega(w)$ by $\Omega(a_{\ell})$ in ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$. Accordingly, $\mathsf{Tab}_{n}$ and $\mathsf{CTab}_{n}$ are considered as multiplicative monoids whose operations are induced by letter encoding; hence, our tableau correspondences are realized as monoid homomorphisms.

With all desired components at our disposal, the tropical linear representation

[TABLE]

of the plactic monoid $\mathcal{P}_{n}$ is obtained by composing the monoid homomorphisms of our main objects (Theorem  7.17), summarized by the diagram

[TABLE]

This representation naturally induces a congruence on $\mathcal{P}_{n}$, and therefore also an equivalence relation on tableaux in $\mathsf{Tab}_{n}$ and $\mathsf{CTab}_{n}$.

In the case of the plactic monoid of rank $3$, $\mathcal{P}_{3}$, the semigroup representation \wp_{3}:\mathcal{P}_{3}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}{\mathfrak{A}^{\times}_{3}}\times{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}} is faithful (Theorem  7.18), more precisely, it is an isomorphism. Consequently, we conclude that $\mathcal{P}_{3}$ admits all the semigroup identities satisfied by the monoid ${\operatorname{TMat}}_{3}(\mathbb{T})$ of $3\times 3$ triangular tropical matrices (Corollary  7.19); in particular, the identities $\Pi_{(C,2,2)}$ in (SID). Furthermore, relying upon the correspondence between reversal of words and transposition of tableaux, standard Young tableaux $\mathsf{STab}_{n}$ are shown to be faithfully realizable by tropical matrices (Theorem 7.20). As tableaux in $\mathsf{STab}_{n}$ bijectively correspond to elements of the symmetric group $S_{n}$, a tropical realization of $S_{n}$ is obtained, linking the theory to Heacke algebras.

Tropical representation theory turns out to be applicable for studying characteristic properties of semigroups in places that the use of classical representation theory is limited. It provides an alternative approach for realization and for the exploration of algebraic-combinatorial objects; especially, their semigroup identities. These identities are of special interest in the theory of semigroup varieties [42], e.g., in finitely generated semigroups of polynomial growth [11, 46]. Applications of the ubiquitous plactic monoid in group representations are well known [10], e.g., in computing products of Schur functions in $n$ variables which are the irreducible polynomial characters of the general linear group $\operatorname{GL}_{n}(\mathbb{C})$, cf. [34]. Tropical linear representations pave the way to studying combinatorial aspects in classical representation theory, including a geometric perspective via projective modules [18, 38]. This paper ease the use of tropical representation by providing a solid bridge between tropical algebra and classical representation theory.

Paper outline

For the reader convenience, this paper is designed as a self-contained document. In  §1, we bring all the relevant definitions and properties of objects to be used in this paper, including basic examples. In §2, we introduce and study the new structure of tropical plactic algebra and its core object, called forward semigroup (Definition 2.2). A brief overview on tropical matrices and their associated digraphs opens  §3, followed by the characterization of tropical corner matrices, providing faithful linear representations of forward semigroups. In §4 we employ corner matrices to construct an explicit troplactic algebra $\mathfrak{A}_{n}$ and to establish the linkage between diagraphs to nondecreasing subwords. The entire  §5 is devoted to the introduction of the cloaktic monoid $\mathcal{K}_{n}$ and the co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$, including their linear representations by $\mathfrak{A}_{n}$ and  ${{}^{\diamond}\joinrel{\mathfrak{A}_{n}}}$, respectively. Young tableaux and configuration tableaux are discussed in  §6, with a special emphasis on numerical functions that allow their mapping to matrices in $\mathfrak{A}_{n}$. Finally, in  §7 we compose our various components to obtain representations and co-representations of configuration tableaux which eventually result as linear representations of the plactic monoid.

To help for a better understanding, our exposition involves many diagrams and examples, including of pathological cases.

1. Preliminaries

To make this paper reasonably self contained, this section, expect parts of §1.3, recalls the relevant notions and terminology, as well as definitions and properties of the algebraic structures to be used in the paper, starting with our special notations.

1.1. Notations

Unless otherwise is specified, the capital letters $I,J$ denote subsets of the neutral numbers $\mathbb{N}:=\{1,2,\dots\}$, while $\mathbb{N}_{0}$ stands for $\{0,1,2,\dots\}$. The finite set $\{1,\dots,n\}$ is often denoted by $N$, for short. By “ordered” we always mean totaly ordered.

Definition 1.1**.**

A subset $J$ of an ordered set $I$ is called convex, written $J\subseteq_{\operatorname{cx}}I$, if for any $i,j\in J$, every $k\in I$ such that $i<k<j$, also belongs to $J$. We write $\{i:j\}$ for the convex subset of $I$ determined by $i\leq j$.

In other words, a convex subset is an “interval”, could be empty or a singleton. For a given $n$, we define $i^{\prime}:=n-i+1$, for $i=1,\dots,n$, ordered now reversely as $n^{\prime}<(n-1)^{\prime}<\cdots<1^{\prime}$. Then $\{j^{\prime}:i^{\prime}\}$ has the same number of elements as $\{i:j\}$ has, and is again convex in $N$.

We write $L_{m}=[\ell_{1},\ell_{2},\dots,\ell_{m}]$ for a sequence of elements $\ell_{t}$ taken from $\{1,\dots,n\}$, where $t=1,\dots,m$. This notation means that the indexing order of elements is preserved. We denote the sequence $[1,\dots,n]$ by $[1:n]$, and write $[i:j]$ for the subsequence $[i,i+1,\dots,j]$ of $[1:n]$.

• •

A subsequence $S$ of $L_{m}$ is notated as $S\sqsubseteq L_{m}$, e.g., $[i:j]\sqsubseteq[1:n]$, for $1\leq i\leq j\leq n$.

• •

\operatorname{len}\big{(}{S}\big{)} denotes the length of $S\sqsubseteq L_{m}$, i.e., the number of its elements.

• •

$S_{k}$ denotes a subsequence of length $k$, $k\geq 0$, where $S_{0}$ stands for the empty sequence.

• •

The notation $\prod_{s\in S_{k}}a_{s}$ stands for the product $a_{s_{1}}\cdot a_{s_{1}}\ {\cdots}\ a_{s_{k}}$, respecting the indexing of the sequence $S_{k}=[s_{1},\dots,s_{k}]$.

• •

A subsequence $S_{k}=[s_{1},s_{2},\dots,s_{k}]$ of length $k$ of $L_{m}$ is

• –

non-decreasing, denoted ${S}^{\uparrow}_{k}\sqsubseteq L_{m}$, if $s_{1}\leq s_{2}\leq\cdots\leq s_{k}$,

• –

increasing if $s_{1}<s_{2}<\cdots<s_{k}$,

• –

non-increasing, denoted ${S}^{\downarrow}_{k}\sqsubseteq L_{m}$, if $s_{1}\geq s_{2}\geq\cdots\geq s_{k}$,

• –

decreasing if $s_{1}>s_{2}>\cdots>s_{k}$.

• •

${S}^{\uparrow}$ and ${S}^{\downarrow}$ denote respectively non-decreasing and non-increasing subsequences of an arbitrary length.

• •

We denote the set of all non-decreasing subsets ${S}^{\uparrow}\sqsubseteq L_{m}$ of $L_{m}$ by ${\operatorname{Sq}}^{\uparrow}(L_{m})$.

• •

A non-decreasing subsequence ${S}^{\uparrow}\sqsubseteq L_{m}$ is said to be maximal in $L_{m}$ if $L_{m}$ has no other non-decreasing subsequence ${R}^{\uparrow}\sqsubseteq L_{m}$ such that ${S}^{\uparrow}\sqsubsetneq{R}^{\uparrow}$. ${\operatorname{MSq}}^{\uparrow}(L_{m})$ denotes the subset of all maximal non-decreasing subsequences of $L_{m}$.

• •

${\operatorname{Sq}}^{\downarrow}(L_{m})$ and ${\operatorname{MSq}}^{\downarrow}(L_{m})$ are defined similarly for non-increasing subsequences.

In this paper we deal only with finite sequences, which are also termed “words” in the context of semigroups.

1.2. Free monoids

A semigroup $\mathcal{S}:=(\mathcal{S},\cdot\,)$ is a set of elements together with an associative binary operation. A  monoid is a semigroup with identity element $e$. Any semigroup $\mathcal{S}$ can be formally adjoined with an identity element  $e$ by declaring that $ea=ae=a$ for all $a\in\mathcal{S},$ so when dealing with multiplicative structures we work with monoids. We write $a^{i}$ for $a\cdot a\cdots a$ with $a\in\mathcal{S}$ repeated $i$ times, and formally identify $a^{0}$ with $e$, when $\mathcal{S}$ is a monoid. An element $o$ of $\mathcal{S}$ is said to be absorbing, usually identified as $\mathbb{0}$, if $oa=ao=o$ for all $a\in\mathcal{S}$. $\mathcal{S}$ is a pointed semigroup if it has an absorbing element $o$.

An Abelian semigroup $\mathcal{S}$ is cancellative with respect to a subset $T\subseteq\mathcal{S}$ if $ac=bc$ implies $a=b$ whenever $a,b\in\mathcal{S}$ and $c\in T$. In this case, $T$ is called a cancellative subset of $\mathcal{S}$, and it generates a subsemigroup in $\mathcal{S}$, also cancellative. Thus, we usually assume that $T$ is a subsemigroup. A semigroup  $\mathcal{S}$ is strictly cancellative if $\mathcal{S}$ is cancellative with respect to itself. The term “congruence” refers to an equivalence relation that respects the operation of its underlying semigroup carrier.

A partially ordered semigroup is a semigroup $\mathcal{S}$ with a partial order $\leq$ that respects the semigroup operation

[TABLE]

for all elements $a,b,c\in\mathcal{S}$. A semigroup $\mathcal{S}$ is ordered if the order $\leq$ is a total order.

We recall some basic definitions from [14], for the reader’s convenience. As customarily, $\mathcal{A}^{*}_{I}$ denotes the free monoid of finite sequences generated by a countably infinite totaly ordered set $\mathcal{A}_{I}:=\{a_{\ell}\ {:}\ \ell\in I\}$, called alphabet, of letters $a_{1},a_{2},a_{3},\dots$ An element $w\in\mathcal{A}^{*}_{I}$ is called a word and, unless it is empty, is written uniquely as

[TABLE]

where $a_{\ell_{t}}\neq a_{\ell_{t+1}}$ for every $t$. We assume that the empty word, denoted $e$, belongs to $\mathcal{A}^{*}_{I}$, serving as the identity. As customarily, $\mathcal{A}^{+}_{I}$ stands for the free sub-semigroup obtained from $\mathcal{A}^{*}_{I}$ by excluding the empty word $e$. When $|I|=n$ is finite, we write $\mathcal{A}_{n}$ for the finite alphabet $\mathcal{A}_{I}=\{a_{1},\dots,a_{n}\}$. A word is read from left to right, to distinguish this reading direction, when needed, we write $\overrightarrow{w}$ for $w$.

The free monoid $\mathcal{A}_{I}^{*}$ is endowed with the familiar lexicographic order, denoted $<_{\operatorname{lx}}$, induced by the order of $\mathcal{A}_{I}$. A sub-alphabet $\mathcal{A}_{J}$ of $\mathcal{A}_{I}$ is said to be convex, written $\mathcal{A}_{I}\subseteq_{\operatorname{cx}}\mathcal{A}_{I}$, if $J\subseteq_{\operatorname{cx}}I$, i.e., $J$ is a convex subset of $I$ (Definition 1.1). We denote by $\mathcal{A}_{[i:j]}$ the convex sub-alphabet $\{a_{i},\dots,a_{j}\}$ of $\mathcal{A}_{n}=\mathcal{A}_{[1:n]}$.

The length of a word $w\in\mathcal{A}^{*}_{I}$ of the form (1.2) is defined (by the standard summation) as $\operatorname{len}(w):=\sum_{t=1}^{m}q_{t}\,.$ Then, since a word $w$ is a finite sequence of letters, $\operatorname{len}(w)$ is well defined and finite, with $\operatorname{len}(w)$ iff $w=e$ is the empty word $e$. A word $w\in\mathcal{A}^{*}_{I}$ is called $k$-uniform if each of its letters appears exactly $k$ times. We say that $w$ is uniform if $w$ is $k$-uniform for some $k$.

A word $u\in\mathcal{A}^{*}_{I}$ is a factor of $w\in\mathcal{A}^{*}_{I}$, written $u|w$, if $w=v_{1}uv_{2}$ for some $v_{1},v_{2}\in\mathcal{A}^{*}_{I}$. When $w=v_{1}v_{2}$, the factors $v_{1}$ and $v_{2}$ are respectively called the prefix and suffix of $w$. If $v_{1}$ is of length $k$ we say that $v_{i}$ is the $k$-prefix of $w$, and denote it by $\operatorname{pre}_{k}(w)$. Similarly, if $\operatorname{len}(v_{2})=k$, we say that $v_{i}$ is the $k$-suffix of $w$, and denote it by $\operatorname{suf}_{k}(w)$.

A word $u$ is a subword of $w$, written $u\subseteq_{\operatorname{wd}}w$, if $w$ can be written as $w=v_{0}u_{1}v_{1}u_{2}v_{2}\cdots u_{m}v_{m}$ where $u_{i}$ and $v_{i}$ are words (possibly empty) such that $u=u_{1}u_{2}\cdots u_{m}$, i.e., the $u_{i}$ are factors of $u$. Clearly, any factor of $w$ is also a subword, but not conversely.

The following notion of $n$-power words was introduced in its full generality in [14, §3.1]. However, for the purpose of the present paper, it suffices to consider a restricted version of this notion, as described below. As seen later, these $n$-power words are the cornerstone of our systematic construction of semigroup identities.

Definition 1.2**.**

Let $C:=C_{m}$ be a finite (nonempty) alphabet, and let $n\in\mathbb{N}$. An $n$-power word $\widetilde{w}:=\widetilde{w}_{(C,p,n)}$ is a nonempty word in $C_{m}^{+}$ such that:

1. (a)

Each letter $a_{\ell}\in C_{m}$ may appear in $\widetilde{w}$ at most $p$-times sequentially, i.e., $a_{\ell}^{q}\nmid\widetilde{w}$ for any $q>p$ and $a_{\ell}\in C_{m}$; 2. (b)

Every word $u\in C^{+}_{m}$ of length $n$ that satisfies rule (a) is a factor of $\widetilde{w}$.

An $n$-power word is uniform if it is uniform as a word.

An $n$-power word needs not be unique in $C^{+}_{m}$, and different $n$-power words may have different length. Often, $n$-power words $\widetilde{w}_{(C,p,n)}$ can be concatenated to a new $n$-power word.

In the present paper, in the view of Theorem 1.8 below, we are mostly interested in the case of $n$-power words over the 2-letter alphabet $C_{2}=\{x,y\}$. As we consider powerwords as generic words, we call the letters $x,y$ variables.

Example 1.3**.**

Suppose $C=\{x,y\}$.

1. (i)

$\widetilde{w}_{(C,2,2)}=yx^{2}y^{2}x$* is a $2$-power uniform word of length $6$.* 2. (ii)

$\widetilde{w}_{(C,3,3)}=xy^{3}xyx^{3}y$* is a $3$-power uniform word of length $10$.*

For more details on $n$-power words see [14, 15].

1.3. Co-mirroring and reversal

Recall that the binary operation of a semigroup is associative.

Definition 1.4**.**

A semigroup homomorphism is a map $\phi:\mathcal{S}\longrightarrow\mathcal{S}^{\prime}$ that preserves the semigroup operation, i.e.,

[TABLE]

for all $a$ and $b$ in $\mathcal{S}$. A monoid homomorphism is a semigroup homomorphism for which $\phi(e)=e^{\prime}$. It is an endomorphism when $\mathcal{S}=\mathcal{S}^{\prime}$.

A presentation of a monoid $\mathcal{M}=(\mathcal{M},\cdot\,)$ (resp. semigroup) is a description of $\mathcal{M}$ in terms of a set of generators $\mathcal{A}_{I}:=\{a_{\ell}\ {|}\ \ell\in I\}$ and a set of relations $\Xi\subset\mathcal{A}_{I}^{*}\times\mathcal{A}_{I}^{*}$ on the free monoid $\mathcal{A}_{I}^{*}$ (resp. on the free semigroup $\mathcal{A}^{+}_{I}$) generated by $\mathcal{A}_{I}$. The monoid $\mathcal{M}$ is then presented as the quotient $\mathcal{A}_{I}^{*}/_{\Xi}$ of the free monoid $\mathcal{A}_{I}^{*}$ by the set of relations $\Xi$, i.e., by a monoid homomorphism

[TABLE]

When $|I|=n$ is finite, we say that $\mathcal{M}$ is finitely presented.

Definition 1.5**.**

The reversal of a word $\overrightarrow{w}=a_{\ell_{1}}a_{\ell_{2}}\cdots a_{\ell_{m}}$ of length $m$ in $\mathcal{A}^{*}_{I}$, $\ell_{i}\in I$, is the word $\overleftarrow{w}=a_{\ell_{m}}a_{\ell_{m-1}}\cdots a_{\ell_{1}}$ of the same length $m$.

The reversal $\overleftarrow{w}$ of a word $\overrightarrow{w}$ is therefore the rewriting of $\overrightarrow{w}$ from right to left, while we formally set $\overleftarrow{e}:=e$. It defines a bijective map

[TABLE]

which is not a monoid homomorphism, since ${\operatorname{rev}}(uv)\neq{\operatorname{rev}}(u){\operatorname{rev}}(v)$. However, ${\operatorname{rev}}(uv)={\operatorname{rev}}(v){\operatorname{rev}}(u)$ for any $u,v\in\mathcal{A}_{I}^{*}$.

Restricting our ground alphabets to finite alphabets, we introduce the following operation:

Definition 1.6**.**

The co-mirror of a letter $a_{\ell}$ in a finite alphabet $\mathcal{A}_{n}$ is defined to be the word

[TABLE]

(The co-mirror of the empty word $e$ is formally set to be $e$.)

The co-mirror of a word $w=a_{\ell_{1}}\cdots a_{\ell_{m}}$ in $\mathcal{A}_{n}^{*}$ is the word defined as

[TABLE]

written ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}(w)$ when $n$ is clear from the context.

Accordingly, for the concatenation $w=uv$ of two words $u,v$ in $\mathcal{A}_{n}^{*}$ we then have

[TABLE]

where $\operatorname{len}({{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}_{n}(w))=(n-1)\operatorname{len}(w)$ for any $w\neq e$. Thus, the co-mirroring of words determines a monoid endomorphism

[TABLE]

with $e\longmapsto e$.

Remark 1.7**.**

The co-mirroring of a finite alphabet $\mathcal{A}_{n}=\{a_{1},\dots,a_{n}\}$ introduces a submonoid in the free monoid $\mathcal{A}_{n}^{*}$, which we denote by ${(\mathcal{A}_{n}^{*})}^{[1]}$, whose generators $a^{\prime}_{\ell}={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}_{n}(a_{\ell})$ are again (totaly) ordered as $a^{\prime}_{1}<\cdots<a^{\prime}_{n}$. In fact, as can be seen form (1.3), this order is just the lexicographic order $<_{\operatorname{lx}}$ of $\mathcal{A}_{n}^{*}$, and thus is compatible with the initial order of  $\mathcal{A}_{n}$.

Applying the co-mirroring map (1.5) inductively, we have a chain of submonoids, a filtration,

[TABLE]

together with the surjective homomorphisms

[TABLE]

Each surjection $\partial_{i}:{(\mathcal{A}_{n}^{*})}^{[i-1]}\;{\relbar\joinrel\twoheadrightarrow}\;{(\mathcal{A}_{n}^{*})}^{[i]}$ is also injective, preserving the lexicographic order of $\mathcal{A}_{n}^{*}$, and thus it is an isomorphism.

Note that $\partial_{i}$ does not necessarily preserve relations nor presentations over $\mathcal{A}_{n}^{*}$.

1.4. Semigroup identities

A (nontrivial) semigroup identity is a formal equality of the form

[TABLE]

where $u$ and $v$ are two different (nonempty) words in the free semigroup $\mathcal{A}^{+}_{I}$, cf. Form (1.2). For a monoid identity, $u$ and $v$ are allowed to be the empty word as well. Therefore, a semigroup identity $\Pi$ determines a single relation $\Pi\in\mathcal{A}^{+}_{I}\times\mathcal{A}^{+}_{I}$ on the free semigroup $\mathcal{A}^{+}_{I}$.

We say that an identity $\Pi:u=v$ is an $n$-variable identity if $u$ and $v$ involve together exactly $n$ letters of $\mathcal{A}_{I}$. An identity $\Pi$ is said to be balanced if the number of occurrences of each letter $a_{i}\in\mathcal{A}_{I}$ is the same in $u$ and in $v$. $\Pi$ is called uniformly balanced if furthermore the words $u$ and $v$ are $k$-uniform for some $k$. The length $\ell(\Pi)$ of $\Pi$ is defined to be $\ell(\Pi):=\max\{\ell(u),\ell(v)\}$, clearly $\ell(u)=\ell(v)$, when $\Pi$ is balanced.

A semigroup $\mathcal{S}:=(\mathcal{S},\cdot\;)$ satisfies the semigroup identity (1.6) if

[TABLE]

Concerning the existence of identities it is suffices to consider $2$-variable identities:

Theorem 1.8** ([14, Theorem 3.10]).**

A semigroup $\mathcal{S}:=(\mathcal{S},\cdot\;)$ that satisfies an $n$-variable identity $\Pi:u=v$, for $n\geq 2$, also satisfies a refined $2$-variable identity $\widehat{\Pi}:\widehat{u}=\widehat{v}$ of exponents $\{1,2\},$ i.e., each letter in $\widehat{u}$ and in $\widehat{v}$ appears sequentially at most twice.

The $n$-power words $\widetilde{w}_{(C,p,n)}$ (Definition 1.2) are utilized to construct a class of nontrivial semigroup identities $\Pi_{(C,p,n)}$. (We use the notation ${\underline{x}}$ and ${\underline{y}}$ to mark a specific instance of the variables $x$ and $y$ in a given expression, although these instances stand for the same variables $x$ and $y$, respectively.)

Construction 1.9**.**

Let $\widetilde{w}_{(C,p,n)}$ be a uniform $n$-power word over $C=\{x,y\}$ such that the words $\ \widetilde{w}_{(C,p,n)}\ {x}\ \widetilde{w}_{(C,p,n)}$ and $\widetilde{w}_{(C,p,n)}\ {y}\ \widetilde{w}_{(C,p,n)}$ are both $n$-power words over $C$. Define the 2-variable balanced identity

[TABLE]

Then, substitute

[TABLE]

to refine (1.7) to the uniformly balanced identity

[TABLE]

where $\widecheck{C}=\{\widecheck{x},\widecheck{y}\}$ and $\widecheck{w}_{(\widecheck{C},p,n)}$ is the word obtained from $\widetilde{w}_{(C,p,n)}$ by substitution (1.8).

Note that $\widecheck{w}_{(\widecheck{C},p,n)}$ needs not be an $n$-power word over $\widecheck{C}=\{\widecheck{x},\widecheck{y}\}$, yet it satisfies law  (a) of Definition 1.2. This summarize Construction 1.9, when a semigroup $\mathcal{S}$ satisfies the identity (1.9), we shortly say that it admits the identity (1.9) by taking $x=uv$ and $y=vu$ for all elements $u,v$ in $\mathcal{S}$.

Example 1.10** ([14, Example 3.9]).**

Let $C=\{x,y\}$.

1. (i)

Using the uniform $2$-power word $\widetilde{w}_{(C,2,2)}=yx^{2}y^{2}x$ as in Example 1.3.(i), we receive the identity

[TABLE] 2. (ii)

Taking the uniform $3$-power word $\widetilde{w}_{(C,3,3)}=xy^{3}xyx^{3}y$ in Example 1.3.(ii), we obtain the identity

[TABLE]

By substituting $x:=\widecheck{x}{\widecheck{y}},$ $y:=\widecheck{y}{\widecheck{x}}$, these identities (1.10) and (1.11) become uniformly balanced.

1.5. Semirings and semimodules

Equipping a set of elements simultaneously by two binary (monoidial) operations, one obtains the following structure.

Definition 1.11**.**

A semiring $R:=(R,+\,,\cdot\;)$ is a set $R$ equipped with two binary operations $+$ and $\cdot\;$, called addition and multiplication, such that:

1. (i)

$(R,+\,)$* is an Abelian monoid with identity element $\mathbb{0}=\mathbb{0}_{R}$;* 2. (ii)

$(R,\cdot\,)$* is a monoid with identity element $\mathbb{1}=\mathbb{1}_{R}$;* 3. (iii)

Multiplication distributes over addition.

A semiring $R$ is called idempotent if $a+a=a$ for every $a\in R$, while it is said to be bipotent (or selective) if $a+b\in\{a,b\}$ for any $a,b\in R$. When the multiplicative monoid $(R,\cdot\,)$ is an abelian group, $R$ is called a semifield.

Clearly, idempotence implies bipotence, but not conversely. A standard general reference for the structure of semirings is [9].

Remark 1.12**.**

Any ordered monoid $(\mathcal{M},\cdot\;)$ gives rise to a semiring, where we define the addition $a+b$ to be $\max\{a,b\}$. Indeed, associativity is clear, and distributivity follows from (1.1). In the same way from an abelian group we obtain a semifield.

Monoid homomorphisms extend naturally to semirings.

Definition 1.13**.**

A homomorphism of semirings is a map $\varphi:R\longrightarrow R^{\prime}$ that preserves addition and multiplication. To wit, $\varphi$ satisfies the following properties for all $a$ and $b$ in $R$:

1. (i)

$\varphi(a+b)=\varphi(a)+\varphi(b)$; 2. (ii)

$\varphi(a\cdot b)=\varphi(a)\cdot\varphi(b)$; 3. (iii)

$\varphi(\mathbb{0}_{R})=\mathbb{0}_{R^{\prime}}$.

A unital semiring homomorphism is a semiring homomorphism that preserves the multiplicative identity, i.e., $\varphi(\mathbb{1}_{R})=\mathbb{1}_{R^{\prime}}$.

In the sequel, unless otherwise is specified, our homomorphisms are assumed to be unital.

In analogy to the case of rings, one defines module over semiring in a straightforward way.

Definition 1.14**.**

An $R$-module $V$ over a semiring $R$ is an additive monoid $(V,+)$ together with a scalar multiplication $R\times V\to V$ satisfying the following properties for all $r_{i}\in R$ and $v,w\in V$:

1. (i)

$r(v+w)=rv+rw;$** 2. (ii)

$(r_{1}+r_{2})v=r_{1}v+r_{2}v;$** 3. (iii)

$(r_{1}r_{2})v=r_{1}(r_{2}v);$** 4. (iv)

$\mathbb{1}_{R}v=v;$** 5. (v)

$\mathbb{0}_{R}v=\mathbb{0}_{V}=r\mathbb{0}_{V}.$**

Modules over semirings are also called semimodules.

2. Tropical plactic algebra

On of the central ground objects of the current paper is the following well-known monoid [30]:

Definition 2.1**.**

The plactic monoid is the monoid $\mathcal{P}_{I}:=\mathsf{PLC}(\mathcal{A}_{I})$, generated by an ordered set of elements $\mathcal{A}_{I}:=\{a_{\ell}\ {|}\ \ell\in I\}$, subject to the equivalence relation $\equiv_{\operatorname{knu}}$ (known as the elementary Knuth relations or the plactic relations) defined by

[TABLE]

for all triplets $a,b,c\in\mathcal{A}_{n}$. Namely, $\mathcal{P}_{I}:=\mathcal{A}_{I}^{*}/_{\equiv_{\operatorname{knu}}}$ with the identity element $e$. When $|I|=n$ is finite, $\mathcal{P}_{I}$ is said to be of rank $n$, or finitely generated, and is denoted by $\mathcal{P}_{n}$. 111In the literature, $\mathsf{PLC}_{I}$ is often assumed to finitely generated, however for large parts of our study are more general and finiteness is not required there.

We write $\mathsf{PLC}_{I}$ for $\mathsf{PLC}(\mathcal{A}_{I})$ when the alphabet $\mathcal{A}_{I}$ is arbitrary. Henceforth, we always assume that $e<a_{\ell}$ for all $a_{\ell}\in\mathcal{P}_{I}$. The relation $\equiv_{\operatorname{knu}}$ is a congruence on the free monoid $\mathcal{A}_{I}^{*}$, e.g. see $\mathcal{A}_{I}^{*}$ [29], which is also denoted by $\equiv_{\operatorname{plc}}$ to indicate its relevance to $\mathsf{PLC}_{I}$.

We aim for an algebra whose multiplicative structure comprises that of the plactic monoid, and in which the Knuth relations are inferred from the algebraic operations. Towards this goal, we begin with an axillary semigroup structure, to be employed for a construction of such a desirable algebra – a special semiring. (In what follows we let $I\subset\mathbb{N}$ be a nonempty subset.)

Definition 2.2**.**

A (partial) forward semigroup $\mathcal{F}_{I}=\langle\,{\mathfrak{f}_{i}\ {|}\ i\in I}\,\rangle$ is a pointed semigroup with an absorbing element $\mathfrak{o}$, generated by the (partial) ordered set of elements 222For simplicity, we assume a countable set of generators, where the generalization to an arbitrary ordered set of generators is obvious. Moreover, one can also generalize this structure by considering a partially ordered set of generators, but such theory is more involved. $\{\mathfrak{f}_{i}\ {|}\ i\in I\}$, subject to the axiom

$\operatorname{FS}$:

$\mathfrak{f}_{j}\mathfrak{f}_{i}=\mathfrak{o}$* whenever $\mathfrak{f}_{j}>\mathfrak{f}_{i}$. *

When $I$ is finite with $|I|=n$, we write $\mathcal{F}_{n}$ for $\mathcal{F}_{I}$.

Note that the semigroup $\mathcal{F}_{I}$ is not assumed to be a (partial) ordered semigroup, i.e., having an order preserving operation $\mathfrak{f}_{i}\geq\mathfrak{f}_{j}\Rightarrow\mathfrak{f}_{i}\mathfrak{f}_{k}\geq\mathfrak{f}_{j}\mathfrak{f}_{k}$ (Definition 1.4), nor a cancellative semigroup, i.e., $\mathfrak{f}_{k}\mathfrak{f}_{i}=\mathfrak{f}_{k}\mathfrak{f}_{j}\not\Rightarrow\mathfrak{f}_{i}=\mathfrak{f}_{j}$. Usually, $\mathcal{F}_{I}$ is non-commutative, since otherwise we would always have $\mathfrak{f}_{i}\mathfrak{f}_{j}=\mathfrak{f}_{j}\mathfrak{f}_{i}=\mathfrak{o}$, implying that $\mathcal{F}_{I}$ consists of $\{\mathfrak{f}_{i}\ {|}\ i\in I\}\cup\{\mathfrak{o}\}.$

Remark 2.3**.**

The elements of a forward semigroup $\mathcal{F}_{I}$ are realized as nondecreasing sequences over $\{\mathfrak{f}_{i}\ {|}\ i\in I\}$. Then it easy to verify that $\mathcal{F}_{I}$ naively admits the Knuth relations (KNT), since in $\mathcal{F}_{I}$ each term of (KNT) equals $\mathfrak{o}$.

Clearly, the forward semigroup $\mathcal{F}_{I}$ is a very degenerated version of the plactic monoid, but useful for the construction of semirings, as seen below in Construction 2.6.

2.1. Tropical plactic algebra

We open this section by introducing a new semiring structure – a key object in this paper – that encapsulates the plactic monoid (cf. Theorem 2.8 below).

Definition 2.4**.**

A tropical plactic algebra $\mathfrak{plc}_{I}$ is a (noncommutative) idempotent semiring $(\mathfrak{plc}_{I},+,\cdot\,)$ with multiplicative identity element $\mathfrak{e}$ and zero $\mathfrak{o}$, generated by an ordered set of elements 22footnotemark: 2 $\{\mathfrak{a}_{i}\ {|}\ i\in I\}$ subject to the axioms (for every $\mathfrak{a}\leq\mathfrak{b}\leq\mathfrak{c}$ in $\{\mathfrak{a}_{i}\ {|}\ i\in I\}$):

$\operatorname{PA1}$:

$\mathfrak{a}=\mathfrak{e}+\mathfrak{a}$;

$\operatorname{PA2}$:

$\mathfrak{b}\mathfrak{a}=\mathfrak{a}+\mathfrak{b}$* when $\mathfrak{b}>\mathfrak{a}$;*

$\operatorname{PA3}$:

$\mathfrak{a}(\mathfrak{b}+\mathfrak{c})=\mathfrak{a}\mathfrak{b}+\mathfrak{c};$**

$\operatorname{PA4}$:

$(\mathfrak{a}+\mathfrak{b})\mathfrak{c}=\mathfrak{a}+\mathfrak{b}\mathfrak{c}.$**

When $|I|=n$ is finite we write $\mathfrak{plc}_{n}$ for $\mathfrak{plc}_{I}$, and say that $\mathfrak{plc}_{n}$ is finitely generated and has rank $n$. (For indexing matter, $\mathfrak{e}$ is also denoted by $\mathfrak{a}_{0}$.)

We abbreviate our terminology and also write troplactic algebra for tropical plactic algebra. Arbitrary elements of $\mathfrak{plc}_{I}$ are denoted by Gothic letters $\mathfrak{u},\mathfrak{v},\mathfrak{a}$, while $\mathfrak{a},\mathfrak{b},\mathfrak{c},$ are devoted for its generators. When it is clear from the context, we suppress the notation of the indexing set $I$ and write $\mathfrak{plc}$ for $\mathfrak{plc}_{I}$, for short.

As $\mathfrak{plc}_{I}$ is an idempotent semiring, then $\mathfrak{u}+\mathfrak{u}=\mathfrak{u}$ for all $\mathfrak{u}\in\mathfrak{plc}_{I}$ (not only for generators $\mathfrak{a}\in\{\mathfrak{a}_{i}\ {|}\ i\in I\}$). While Axioms $\operatorname{PA1}$ and $\operatorname{PA2}$ are easily figured out, Axioms $\operatorname{PA3}$ and $\operatorname{PA4}$ are more complicated. To better understand them, they are illustrated by the diagrams:

[TABLE]

[TABLE]

in which the columns stand for products of the letters $\mathfrak{a},\mathfrak{b}$ and $\mathfrak{c}$ and the rows for the additive terms.

Remark 2.5**.**

Given a product $\mathfrak{u}=\mathfrak{a}_{\ell_{1}}\cdots\mathfrak{a}_{\ell_{m}}$ in $\mathfrak{plc}_{I}$, with $\ell_{1},\dots,\ell_{m}\in I$, we write $\mathfrak{u}^{\dagger}$ for the realization of $\mathfrak{u}$ as a word in the free semigroup $\mathcal{A}^{+}_{I}$ over the set of symbols $\mathcal{A}_{I}=\{\mathfrak{a}_{i}\ {|}\ i\in I\}$. Then $\mathcal{A}_{I}^{+}$ – the multiplicative structure of $\mathfrak{plc}_{I}$ – is equipped with a lexicographic order, determined by order of the symbol set $\mathcal{A}_{I}$. We denote this lexicographic order by $<_{\operatorname{lx}}.$

Troplactic algebras appear naturally as semimodules over a semiring (Definition  1.14), often as a projective semimodule realized as projective matrices, where then the equalities in axioms $\operatorname{PA1}$–$\operatorname{PA4}$ are taken with respect to the appropriate setting. A particular abstract construction of plactic algebra can be obtained by the use of forward semigroups (Definition 2.2).

Construction 2.6**.**

Let $\mathcal{F}_{I}=\langle\,{\mathfrak{f}_{i}\ {|}\ i\in I}\,\rangle$ be a forward semigroup and $\mathfrak{d}$ an adjoint distinguished additively idempotent element $\mathfrak{d}=\mathfrak{d}+\mathfrak{d}$ that commutes with each element $\mathfrak{f}_{i}$, i.e., $\mathfrak{d}\mathfrak{f}_{i}=\mathfrak{f}_{i}\mathfrak{d}$. Set $\mathfrak{a}_{i}:=\mathfrak{d}+\mathfrak{f}_{i}$, whose order is induced by the order $\mathcal{F}_{I}$, i.e.,

[TABLE]

to define a troplactic algebra $\mathfrak{plc}_{I}$ as the algebra generated by $\mathfrak{a}_{i}$. We often take $\mathfrak{d}=\mathfrak{e}$ to be the multiplicative identity of $\mathfrak{plc}_{I}$.

In this setup, if $\mathfrak{a}_{i}<\mathfrak{a}_{j}$, then $\mathfrak{a}_{j}\mathfrak{a}_{i}=\mathfrak{d}(\mathfrak{a}_{i}+\mathfrak{a}_{j})$. Indeed, $\mathfrak{a}_{j}\mathfrak{a}_{i}=(\mathfrak{d}+\mathfrak{f}_{j})(\mathfrak{d}+\mathfrak{f}_{i})=\mathfrak{d}^{2}+\mathfrak{d}\mathfrak{f}_{i}+\mathfrak{d}\mathfrak{f}_{j}=\mathfrak{d}(\mathfrak{d}+\mathfrak{f}_{i}+\mathfrak{f}_{j})=\mathfrak{d}(\mathfrak{a}_{i}+\mathfrak{a}_{j}).$ Thus, for the case $\mathfrak{d}=\mathfrak{e}$, only axioms $\operatorname{PA3}$ and $\operatorname{PA4}$ are enforced, while Axiom $\operatorname{PA1}$ is obtained by construction and Axiom $\operatorname{PA2}$ follows from the structure of $\mathcal{F}_{I}$.

Construction 2.6 turns out to be very useful in applications, especially for troplactic matrix algebra, as its arithmetics provides a natural additive decomposition for multiplicative expressions, followed from Axiom $\operatorname{PA2}$. (Note that $\mathfrak{plc}_{I}$ is not a bipotent semiring and its addition is not necessarily determined as maximum.)

Remark 2.7**.**

Let $\mathfrak{a},\mathfrak{b},\mathfrak{c}\in\{\mathfrak{a}_{i}\ {|}\ i\in I\}$ be the generators of $\mathfrak{plc}_{I}$, and assume that $\mathfrak{a}\leq\mathfrak{b}\leq\mathfrak{c}$.

1. (i)

Declaring that $\mathfrak{e}<\mathfrak{a}_{i}$ for all $i\in I$, Axiom $\operatorname{PA1}$ is then derived directly from Axiom $\operatorname{PA2}$. Furthermore, in this case, the equality $\mathfrak{e}=\mathfrak{a}+\mathfrak{e}$ implies that $\mathfrak{a}=\mathfrak{e}$. Indeed $\mathfrak{e}=\mathfrak{a}+\mathfrak{e}=\mathfrak{a}\mathfrak{e}=\mathfrak{a}$. 2. (ii)

The multiplicative identity $\mathfrak{e}$ is also an additive identity (by Axiom $\operatorname{PA1}$)

[TABLE]

and in particular it is idempotent with respect to both addition and multiplication, i.e.,

[TABLE] 3. (iii)

Axiom $\operatorname{PA1}$ implies

[TABLE]

since $\mathfrak{a}\mathfrak{b}=\mathfrak{a}(\mathfrak{b}+\mathfrak{e})=\mathfrak{a}\mathfrak{b}+\mathfrak{a}$, and $\mathfrak{a}\mathfrak{b}=(\mathfrak{a}+\mathfrak{e})\mathfrak{b}=\mathfrak{a}\mathfrak{b}+\mathfrak{b}.$ Therefore, by idempotence of $\mathfrak{u}=\mathfrak{a}\mathfrak{b}$,

[TABLE]

which implies that

[TABLE]

and furthermore that

[TABLE]

Inductively, together with the use of Axiom $\operatorname{PA2}$, for an arbitrary product $\mathfrak{a}_{\ell_{1}}\cdots\mathfrak{a}_{\ell_{m}}$ we obtain

[TABLE]

Namely, a decomposition of products to sums of non-decreasing terms. 4. (iv)

When $\mathfrak{a}=\mathfrak{b},$ Axiom $\operatorname{PA3}$ reads as $\mathfrak{a}\mathfrak{a}+\mathfrak{a}\mathfrak{c}=\mathfrak{a}\mathfrak{a}+\mathfrak{c},$ and thus by (iii) we have

[TABLE]

hence $(\mathfrak{a}+\mathfrak{c})^{2}=\mathfrak{a}^{2}+\mathfrak{c}^{2}$. 5. (v)

When $\mathfrak{b}=\mathfrak{c},$ Axiom $\operatorname{PA4}$ reads as $\mathfrak{b}\mathfrak{b}+\mathfrak{a}\mathfrak{b}=\mathfrak{b}\mathfrak{b}+\mathfrak{a}.$ 6. (vi)

If $\mathfrak{a}\mathfrak{b}=\mathfrak{e}$, then, by Axioms $\operatorname{PA1}$, $\operatorname{PA2}$, and part (iii), we have

[TABLE]

and thus also $\mathfrak{b}\mathfrak{a}=\mathfrak{e}$. Accordingly, when $\mathfrak{a}\mathfrak{b}=\mathfrak{e}$ for all $\mathfrak{b}>\mathfrak{a}$, all the elements of $\mathfrak{plc}_{I}$ are of the form $\mathfrak{a}_{\ell}^{k}$, with $k\in\mathbb{N}.$

The main virtue of the troplactic algebra is that the (multiplicative) Knuth relations can be deduced from its semiring structure. Therefore, multiplicatively, $\mathfrak{plc}_{I}$ has the structure of the plactic monoid  $\mathcal{P}_{I}$ (Definition 2.1), which in general is not degenerated.

Theorem 2.8**.**

The troplactic algebra $\mathfrak{plc}_{I}$ admits the Knuth relations (KNT):

[TABLE]

where $\mathfrak{a},\,\mathfrak{b},\,\mathfrak{c}\in\{\mathfrak{a}_{i}\ {|}\ i\in I\}$, and thus mutiplictively forms a plactic monoid.

Proof.

We employ Axiom $\operatorname{PA1}$ via Remark 2.7.(ii).

1. $\operatorname{KN1}$:

By axiom $\operatorname{PA2}$:

[TABLE]

while Axiom $\operatorname{PA1}$  gives (cf. Remark 2.7.(iii))

[TABLE]

Composing both by using Axiom $\operatorname{PA3}$, we obtain

[TABLE]

as desired. 2. $\operatorname{KN2}$:

By axiom $\operatorname{PA4}$

[TABLE]

On the other hand, Axiom $\operatorname{PA1}$  and Axiom $\operatorname{PA2}$ imply

[TABLE]

yielding the relation $\mathfrak{b}\mathfrak{a}\mathfrak{c}=\mathfrak{b}\mathfrak{c}\mathfrak{a}$.

Thus, $\mathfrak{plc}_{I}$ admits the Knuth relations (KNT), and therefore the congruence $\equiv_{\operatorname{knu}}$. ∎

Theorem 2.8 also implies that one can produce a troplactic algebra form a given plactic monoid $\mathcal{P}_{I}=\mathcal{A}_{I}/_{\equiv_{\operatorname{knu}}}$ (Definition 2.1) by adjoining a zero element and introducing a formal additive (commutative) operation $+$, subject to Axioms $\operatorname{PA2}$-$\operatorname{PA4}$.

Example 2.9**.**

**

1. (i)

$\mathfrak{plc}_{1}$* is the trivial troplactic algebra has rank $1$ and it is generated by a single, and thus $\mathfrak{plc}_{1}=\{\mathfrak{a}^{k}\ {|}\ k\in\mathbb{N}_{0}\}.$* 2. (ii)

The troplactic algebra $\mathfrak{plc}_{2}$ with two generators $\mathfrak{b}>\mathfrak{a}$ consists of the elements of the forms

[TABLE]

where $i,j,p,q\in\mathbb{N}_{0}$. If $\mathfrak{a}\mathfrak{b}=\mathfrak{e}$, then also $\mathfrak{b}\mathfrak{a}=\mathfrak{e}$ by Remark 2.7.(vi), and $\mathfrak{plc}_{2}$ contains only powers of $\mathfrak{a}$ and $\mathfrak{b}$.

In troplactic algebra, by the proof of Theorem 2.8, we see that the terms of the form $\mathfrak{a}\mathfrak{c}\mathfrak{b}=\mathfrak{c}\mathfrak{a}\mathfrak{b}=\mathfrak{a}\mathfrak{b}+\mathfrak{c}$ and $\mathfrak{b}\mathfrak{a}\mathfrak{c}=\mathfrak{b}\mathfrak{c}\mathfrak{a}=\mathfrak{a}+\mathfrak{b}\mathfrak{c}$ decompose in $\mathfrak{plc}_{I}$ to sums of products of length $1$ and $2$. On the other hand $\mathfrak{c}\mathfrak{b}\mathfrak{a}=\mathfrak{a}+\mathfrak{b}+\mathfrak{c}$ decomposes to a sum of generators, while $\mathfrak{a}\mathfrak{b}\mathfrak{c}$ is not necessarily decomposable. Thus, in general, the products $\mathfrak{a}\mathfrak{b}\mathfrak{c}$ and $\mathfrak{c}\mathfrak{b}\mathfrak{a}$ in $\mathfrak{plc}_{I}$ are different from $\mathfrak{a}\mathfrak{c}\mathfrak{b}=\mathfrak{c}\mathfrak{a}\mathfrak{b}$ and $\mathfrak{b}\mathfrak{a}\mathfrak{c}=\mathfrak{b}\mathfrak{c}\mathfrak{a}$, and multiplicatily $\mathfrak{plc}_{I}$ has a non-degenerated plactic structure.

In the forthcoming study of troplactic algebra, we intensively use our notations for sequences, cf. §1.1.

Theorem 2.10**.**

Suppose $L_{m}=[\ell_{1},\dots,\ell_{m}]$ is nondecreasing, with $\ell_{1},\dots,\ell_{m}\in I$ and $m\geq 1$, then

[TABLE]

Note that, as $L_{m}$ is assumed nondecreasing, this means that we only care about maximal nondecreasing sequences and ignore all shorter subsequences. That is, there are no nested nondecreasing sequences.

Proof.

Proof by induction on $m$. The case of $m=1$ is trivial, and we start $m=2$. Let $L_{2}=[i,j]$, where $i=\ell_{1}\leq\ell_{2}=j$. Then, by Remark 2.7 we have

[TABLE]

Assuming the implication for $m-1$, since $L_{m}$ assumed nondecreasing, we have

[TABLE]

as required. ∎

Remark 2.11**.**

From Theorem 2.10 we learn that for a nondecreasing sequence (i.e., word) the addition of $\mathfrak{plc}_{I}$ is compatible with the lexicographic order (cf. Remark 2.5),

[TABLE]

Note that in general $\mathfrak{a}<\mathfrak{b}$ does not imply $\mathfrak{a}+\mathfrak{b}=\mathfrak{b}$, and also that the stand alone condition $\mathfrak{u}^{\dagger}\leq_{\operatorname{lx}}\mathfrak{v}^{\dagger}$ does not imply $\mathfrak{u}+\mathfrak{v}=\mathfrak{v}$.

The linkage between inclusion of words to their lexicographic order allow to consider maximality in a suitable algebraic context.

Corollary 2.12**.**

Any product $\mathfrak{a}_{\ell_{1}}\cdots\mathfrak{a}_{\ell_{m}}$ in ${\operatorname{plc}}_{I}$, with $\ell_{1},\dots,\ell_{m}\in I$ and $m\geq 1$, decomposes as

[TABLE]

Proof.

If $S\sqsubseteq L_{m}$ is not nondecreasing then $\prod_{s\in S}\mathfrak{a}_{s}$ splits to a sum of terms by Axiom $\operatorname{PA2}$, otherwise for the nondecreasing sequences ${S}^{\uparrow}\sqsubseteq L_{m}$ it is enough to restrict to the maximal ones, by Theorem  2.10 and Remark 2.11. ∎

In particular cases of strictly decreasing words, this Corollary provides us the following useful computational simplification.

Corollary 2.13**.**

Any finite product $(\mathfrak{a}_{\ell_{1}})^{q_{1}}\cdots(\mathfrak{a}_{\ell_{m}})^{q_{m}}$ in ${\operatorname{plc}}_{I}$ with $\mathfrak{a}_{\ell_{1}}>\mathfrak{a}_{\ell_{2}}>\cdots>\mathfrak{a}_{\ell_{m}}$ decomposes to

[TABLE]

and thus

[TABLE]

We also have the following generalization to arbitrary finite products.

Corollary 2.14**.**

$\mathfrak{u}^{m}+\mathfrak{u}^{m-1}+\cdots+\mathfrak{u}=\mathfrak{u}^{m}$* for every finite product $\mathfrak{u}\in\mathfrak{plc}_{I}$, i.e., a word $\mathfrak{u}^{\dagger}$ over $\mathcal{A}_{I}$.*

Proof.

For every $k<m$, each term in decomposition (2.4) of $\mathfrak{u}^{k}$ is also contained in one of the terms of decomposition (2.4) of $\mathfrak{u}^{m}$, both are nondecreasing, and we are done by Theorem 2.10. ∎

Recall that the elementary symmetric polynomials in $n$ variables $X_{1},\dots,X_{n}$ are defined as

[TABLE]

for $k=1,\dots,n$. In analogy, the elementary symmetric expressions on a finite alphabet $\mathcal{A}_{n}$ are syntactic formulas, defined as

[TABLE]

for $k=1,\dots,n$. By Remark 2.7.(iii), for expressions taken over $\mathfrak{plc}_{n}$, we have

[TABLE]

for any $\ell\geq k$.

Lemma 2.15**.**

Any pair of generators $\mathfrak{a}$ and $\mathfrak{b}$ of $\mathfrak{plc}_{I}$ admits the Frobenius property:

[TABLE]

Proof.

Proof by induction on $m$. The case of $m=2$ is given by Remark 2.7.(iv):

[TABLE]

Assuming the implication holds for $m-1$, write

[TABLE]

as desired. ∎

As an immediate consequence of this proof we obtain the following.

Corollary 2.16**.**

Suppose $\mathfrak{a}<\mathfrak{b}$ are generators of $\mathfrak{plc}_{I}$, then

[TABLE]

for any $m\in\mathbb{N}.$

The troplactic algebra $\mathfrak{plc}_{I}$ in general does not admit the Frobenius property (2.8), i.e., for elements that are not generators; for example

[TABLE]

since $\mathfrak{a}\mathfrak{b}\mathfrak{c}$ appear in $\mathfrak{u}$ but not in $\mathfrak{v}.$

Remark 2.17**.**

Not all the elements of $\mathfrak{plc}_{I}$ can be rewritten as products, i.e., $\mathfrak{plc}_{I}$ has elements that are not multiplicatively generated. For example, the element $ad+bc$ is not provided as any word $u^{\dagger}$ in $\mathfrak{plc}_{I}$. Indeed, otherwise the terms $ad$ and $bc$ would be subwords of $u^{\dagger}$, which does not contain other increasing subwords of length 2, but this impossible.

2.2. Dual tropical plactic algebra

We turn to a dual version of a troplactic algebra $\mathfrak{plc}_{I}$ (cf. Definition  2.4), to be employed later, carrying analogous properties.

Definition 2.18**.**

A dual troplactic algebra, denoted $\mathfrak{plc}^{\wedge}_{I}$, is a (noncommutative) idempotent semiring $(\mathfrak{plc}^{\wedge}_{I},\,\widehat{+}\,,\cdot\,)$ with (multiplicative) identity element $\mathfrak{e}$ and zero $\mathfrak{o}$, generated by an ordered set of elements $\{\mathfrak{a}_{i}\ {|}\ i\in I\}$ subject to the axioms (for every $\mathfrak{a}\leq\mathfrak{b}\leq\mathfrak{c}$ in $\{\mathfrak{a}_{i}\ {|}\ i\in I\}$):

$\operatorname{PA1}^{\prime}$:

$\mathfrak{a}=\mathfrak{e}\,\widehat{+}\,\mathfrak{a}$;

$\operatorname{PA2}^{\prime}$:

$\mathfrak{a}\mathfrak{b}=\mathfrak{a}\,\widehat{+}\,\mathfrak{b}$* when $\mathfrak{b}>\mathfrak{a}$;*

$\operatorname{PA3}^{\prime}$:

$(\mathfrak{b}\,\widehat{+}\,\mathfrak{c})\mathfrak{a}=\mathfrak{b}\mathfrak{a}\,\widehat{+}\,\mathfrak{c};$**

$\operatorname{PA4}^{\prime}$:

$\mathfrak{c}(\mathfrak{b}\,\widehat{+}\,\mathfrak{a})=\mathfrak{a}\,\widehat{+}\,\mathfrak{c}\mathfrak{b}.$**

For indexing matter, we denote $\mathfrak{e}$ also by $\mathfrak{a}_{0}$.

(We use the special notation “$\,\widehat{+}\,$” for the addition in $\mathfrak{plc}^{\wedge}_{I}$ to distinguish it from the addition of $\mathfrak{plc}_{I}$.)

In analogy to Remark 2.7, by changing nondecreasing sequences by nonincreasing sequences, we now have following properties.

Remark 2.19**.**

Let $\mathfrak{a}\leq\mathfrak{b}\leq\mathfrak{c}$ be elements in the generating set $\{\mathfrak{a}_{i}\ {|}\ i\in I\}$ of $\mathfrak{plc}^{\wedge}_{I}$.

1. (i)

Axiom $\operatorname{PA1}^{\prime}$ is derived directly from Axiom $\operatorname{PA2}^{\prime}$ by declaring that $\mathfrak{e}<\mathfrak{a}_{i}$ for all $i$. 2. (ii)

The multiplicative identity $\mathfrak{e}$ is also an additive identity (by Axiom $\operatorname{PA1}^{\prime}$)

[TABLE] 3. (iii)

Since $\mathfrak{a}\mathfrak{b}=\mathfrak{a}(\mathfrak{b}\,\widehat{+}\,\mathfrak{e})=\mathfrak{a}\mathfrak{b}\,\widehat{+}\,\mathfrak{a}$, and $\mathfrak{a}\mathfrak{b}=(\mathfrak{a}\,\widehat{+}\,\mathfrak{e})\mathfrak{b}=\mathfrak{a}\mathfrak{b}\,\widehat{+}\,\mathfrak{b}$, by Axiom $\operatorname{PA1}^{\prime}$, the idempotence of $\mathfrak{plc}^{\wedge}_{I}$ implies that

[TABLE]

This gives

[TABLE]

while using Axiom $\operatorname{PA2}^{\prime}$ inductively for an arbitrary product $\mathfrak{a}_{\ell_{1}}\cdots\mathfrak{a}_{\ell_{m}}$ we get

[TABLE] 4. (iv)

When $\mathfrak{a}=\mathfrak{b},$ Axiom $\operatorname{PA3}^{\prime}$ reads $\mathfrak{a}\mathfrak{a}\,\widehat{+}\,\mathfrak{c}\mathfrak{a}=\mathfrak{a}\mathfrak{a}\,\widehat{+}\,\mathfrak{c},$ and thus by (iii) we have

[TABLE]

and thus $(\mathfrak{a}\,\widehat{+}\,\mathfrak{c})^{2}=\mathfrak{a}^{2}\,\widehat{+}\,\mathfrak{c}^{2}$. 5. (v)

When $\mathfrak{b}=\mathfrak{c},$ Axiom $\operatorname{PA4}^{\prime}$ reads $\mathfrak{b}\mathfrak{b}\,\widehat{+}\,\mathfrak{b}\mathfrak{a}=\mathfrak{a}\,\widehat{+}\,\mathfrak{b}\mathfrak{b}.$ 6. (vi)

If $\mathfrak{b}\mathfrak{a}=\mathfrak{e}$, then, by Axioms $\operatorname{PA1}^{\prime}$, $\operatorname{PA2}^{\prime}$, and part (iii), we have

[TABLE]

and thus also $\mathfrak{a}\mathfrak{b}=\mathfrak{e}$. This implies that all the elements of $\mathfrak{plc}_{I}$ are of the form $\mathfrak{a}_{\ell}^{k}$, with $k\in\mathbb{N}.$

Theorem 2.20**.**

The dual troplactic algebra $\mathfrak{plc}^{\wedge}_{I}$ admits the Knuth relations (KNT)

[TABLE]

where $\mathfrak{a},\;\mathfrak{b},\;\mathfrak{c}\in\{\mathfrak{a}_{i}\ {|}\ i\in I\}$.

Proof.

We fellow similar arguments as in the proof of Theorem 2.8.

1. $\operatorname{KN1}$:

By axiom $\operatorname{PA4}^{\prime}$

[TABLE]

On the other hand Axiom $\operatorname{PA1}^{\prime}$  and Axiom $\operatorname{PA2}^{\prime}$ imply

[TABLE]

yielding $\mathfrak{c}\mathfrak{a}\mathfrak{b}=\mathfrak{a}\mathfrak{c}\mathfrak{b}$. 2. $\operatorname{KN2}$:

By axiom $\operatorname{PA2}^{\prime}$:

[TABLE]

while Axiom $\operatorname{PA1}^{\prime}$  gives

[TABLE]

Composing both by using Axiom $\operatorname{PA3}^{\prime}$, we obtain

[TABLE]

as required.

Thus, $\mathfrak{plc}^{\wedge}_{n}$ admits the Knuth relations (KNT), and therefore the congruence $\equiv_{\operatorname{knu}}$. ∎

With similar proofs, adapted to Definition 2.18, the results of §2.1 also hold for dual troplactic algebras, replacing nondecreasing sequences by nonincreasing sequences.

3. Tropical matrices

The tropical max-plus semiring $(\mathbb{T},\vee,\cdot\,)$ is the set $\mathbb{T}:=\mathbb{R}\cup\{-\infty\}$ endowed with the operations of maximum and summation (written in the standard algebraic way),

[TABLE]

serving respectively as the semiring addition and multiplication [40]. (We save the sign $``+"$ for the standard summation, and use “$\vee$” for addition.) This semiring is additively idempotent, i.e., $a\vee a=a$ for every $a\in\mathbb{T}$, in which $\mathbb{0}:=-\infty$ is the zero element and $\mathbb{1}:=0$ is the multiplicative unit. (Equivalently one can set $\mathbb{T}:=\mathbb{N}\cup\{-\infty\}$, $\mathbb{T}:=\mathbb{Z}\cup\{-\infty\}$, or $\mathbb{T}:=\mathbb{Q}\cup\{-\infty\}$).

Dually, the min-plus semiring $(\mathbb{T}_{\wedge},\wedge,\cdot\,)$ is the set $\mathbb{T}_{\wedge}:=\mathbb{R}\cup\{\infty\}$ equipped with the operations

[TABLE]

addition and multiplication respectively. The addition $\wedge$ over $\mathbb{R}$ can be written in terms of $\vee$ as $\displaystyle{a\wedge b=\frac{ab}{a\vee b},}$ or as

[TABLE]

where $-a$ stands the standard negation of $a\in\mathbb{R}$ – the tropical multiplicative inverse of $a$. (When $a=-\infty$ in (3.1) we implicitly take $-a=\infty$.) We write $\mathbb{T}$ for the max-plus semiring $(\mathbb{T},\vee,\cdot\,)$ and $\mathbb{T}_{\wedge}$ for the min-plus semiring $(\mathbb{T},\wedge,\cdot\,)$.

The boolean semiring is the two element idempotent semiring $\mathbb{B}:=(\{0,1\},\vee,\cdot\,),$ whose addition and multiplication are given for $a,b\in\{0,1\}$ by:

[TABLE]

$\mathbb{B}$ embeds naturally in $\mathbb{T}$ via $0\longmapsto\mathbb{0}$ and $1\longmapsto\mathbb{1}$, while the projection

[TABLE]

is a semiring homomorphism (Definition 1.13).

3.1. Tropical matrices

Tropical matrices are matrices with entries in $\mathbb{T}$, whose multiplication is induced from the operations of $\mathbb{T}$ as in the familiar matrix construction. The set of all $n\times n$ tropical matrices forms a noncommutative semiring, denoted by ${\operatorname{Mat}}_{n}(\mathbb{T})$. The unit $I$ of ${\operatorname{Mat}}_{n}(\mathbb{T})$ is the matrix with $\mathbb{1}:=0$ on the main diagonal and whose off-diagonal entries are all $\mathbb{0}:=-\infty$. The zero matrix is denoted by $(\mathbb{0})$, the absorbing element of ${\operatorname{Mat}}_{n}(\mathbb{T})$. Formally, for any nonzero matrix $A\in{\operatorname{Mat}}_{n}(\mathbb{T})$ we set $A^{0}:=I$. A matrix $A\in{\operatorname{Mat}}_{n}(\mathbb{T})$ with entries $a_{i,j}$, $i,j=1,\dots,n$, is written as $A=(a_{i,j})$. For a pair of matrices $A,B\in{\operatorname{Mat}}_{n}(\mathbb{T})$, we define

[TABLE]

and $A>B$ if $A\geq B$ with $a_{i,j}>b_{i,j}$ for some $i,j$. Equivalently, we have

[TABLE]

Therefore, in general, ${\operatorname{Mat}}_{n}(\mathbb{T})$ is a partially ordered semiring with $(\mathbb{0})<A$ for all $A$. We denote the monoid of all upper tropical triangular matrices by ${\operatorname{TMat}}_{n}(\mathbb{T})$.

Remark 3.1**.**

Given a matrix $A=(a_{i,j})$ in ${\operatorname{Mat}}_{n}(\mathbb{T})$ we write $-A$ for the matrix

[TABLE]

Then, by (3.1), the entry-wise operation gives

[TABLE]

i.e., the entry-wise minimum of the matrices $A$ and $B$.

The structure map is the onto monoid homomorphism

[TABLE]

sending a tropical matrix $A\in{\operatorname{Mat}}_{n}(\mathbb{T})$ to the boolean matrix $\widetilde{\pi}(A)\in{\operatorname{Mat}}_{n}(\mathbb{B})$ defined as $\widetilde{\pi}(A)=(\pi(a_{i,j}))$, by the entry-wise mapping (3.2). (It is a “value forgetful” homomorphism that maps nonzero entries to  $1$ and zero-entries to  [math].) The matrix $\widetilde{\pi}(A)$ is called the structure matrix of $A$.

Tropical matrix algebra was systematically studied in its generalized context of supertropical algebra [20], providing many tropical analogues to classical results [13, 21, 22, 23], while subgroup and semigroups structures of these matrices were discussed in [38, 48]. As a matrix monoid, in comparison to matrices over a field, important submonmoids of ${\operatorname{Mat}}_{n}(\mathbb{T})$ have a very special behavior, as have been dealt in [14, 15, 19], yielding the following results.

Theorem 3.2** ([14, Theorem 4.11]).**

The submonoid ${\operatorname{TMat}}_{n}(\mathbb{T})\subset{\operatorname{Mat}}_{n}(\mathbb{T})$ of all upper (or dually lower) triangular tropical matrices satisfies the semigroup identities defined in Construction 1.9:

[TABLE]

for $C=\{x,y\}$, with $x=AB$ and $y=BA$.

Theorem 3.3** ([15, Theorem 4.14]).**

Any nonsingular subsemigroup $\mathcal{M}_{n}\subset{\operatorname{Mat}}_{n}(\mathbb{T})$ satisfies the identities   (3.6) by letting $x=A^{n!}B^{n!}$ and $y=B^{n!}A^{n!}$.

Clearly, these results hold also to boolean matrices.

We recall some basic definitions from [12, 13, 20, 21].

Definition 3.4**.**

Given a tropical matrix $A=(a_{i,j})$ in ${\operatorname{Mat}}_{n}(\mathbb{T})$.

1. (i)

The transpose of $A$ is defined as $A^{\operatorname{t}}=(a_{j,i})$ and satisfies $(AB)^{\operatorname{t}}=B^{\operatorname{t}}A^{\operatorname{t}}$ for any $B\in{\operatorname{Mat}}_{n}(\mathbb{T})$. 2. (ii)

The permanent of $A$ is defined as:

[TABLE]

where $\mathfrak{S}_{n}$ denotes the set of all the permutations over $\{1,\dots,n\}$. 3. (iii)

$A$* is called nonsingular if there exist a unique permutation $\tau_{A}\in\mathfrak{S}_{n}$ that reaches ${\operatorname{per}}(A)$; that is, ${\operatorname{per}}(A)=\prod_{i}a_{i,\tau_{A}(i)}\;.$ Otherwise, $A$ is called singular.* 4. (iv)

The (tropical) rank of $A$ is the largest $k$ for which $A$ has a $k\times k$ nonsingular sub-matrix. (Equivalently, the rank is the maximal number of independent columns (or rows) of $A$, cf. **[20]**.) 5. (v)

The trace and the multiplicative trace of $A$ are defined respectively as

[TABLE]

It is easily seen that ${\operatorname{per}}(A)\geq\operatorname{mtr}(A)$ and $\operatorname{mtr}(AB)\geq\operatorname{mtr}(A)\operatorname{mtr}(B)$, for any matrices $A,B\in{\operatorname{Mat}}_{n}(\mathbb{T})$.

3.2. Digraph view to tropical matrices

Square tropical matrices correspond uniquely to weighted diagraphs, whose products interpreted as weights of paths [2], as described below. An $n\times n$ tropical matrix $A=(a_{i,j})$ is associated to the weighted digraph $G_{A}:=(\mathcal{V},\mathcal{E})$ on the vertex set $\mathcal{V}:=\{1,\dots,n\}$ with a directed edge $\varepsilon_{i,j}:=(i,j)\in\mathcal{E}$ of weight $a_{i,j}$ from $i$ to $j$ for every $a_{i,j}\neq\mathbb{0}$. The digraph of the transpose matrix $A^{\operatorname{t}}$ is obtained by redirecting the edges of $G_{A}$, i.e., an edge $(i,j)$ is replaced by the edge $(j,i)$ of the same weight.

A path $\gamma$ is a sequence of edges $\varepsilon_{i_{1},j_{1}},\dots,\varepsilon_{i_{m},j_{m}}$, with $j_{k}=i_{k+1}$ for every $k=1,\dots,m-1$. We write $\gamma:=\gamma_{i,j}$ to indicate that $\gamma$ is a path from $i=i_{1}$ to $j=j_{m}$. The length $\operatorname{len}(\gamma)$ of a path $\gamma$ is the number of its edges. Formally, we consider also paths of length [math], which we call empty paths. The weight $\omega(\gamma)$ of a path $\gamma$ is the tropical product of the weights of all the edges $\varepsilon_{i_{k},j_{k}}$ composing $\gamma$, counting repeated edges. The weight of an empty path is formally set to be $\mathbb{0}$.

A path is simple if it crosses each vertex at most once – it does not have repeating vertices. (Accordingly, an empty path is considered also as simple.) A (simple) path that starts and ends at the same vertex is called a (simple) cycle. An edge $\rho_{i}:=\varepsilon_{i,i}$ is called a loop. We write $(\rho)^{k}$ for the composition $\rho\circ\dots\circ\rho$ of a loop $\rho$ repeated $k$ times, and call it a multiloop. The notation $(\rho)^{0}$ is formal, stands for the empty loop, which can be realized as a vertex. A graph is acyclic if it has no cycles of length $>1$ (i.e., it may have loops).

In combinatorial view, entries in powers of a tropical matrix $A\in{\operatorname{Mat}}_{n}(\mathbb{T})$ correspond to paths of maximal weights in the associated digraph $G_{A}$ of $A$. Namely, the $(i,j)$-entry of the matrix $A^{m}$ corresponds to the highest weight over all the paths $\gamma_{i,j}$ from $i$ to $j$ of length $m$ in $G_{A}$.

The graph view of products $A_{\ell_{1}}\cdots A_{\ell_{m}}$ of different $n\times n$ matrices $A_{\ell_{t}}\in\{A_{\ell}\ {|}\ \ell\in I\}$ is more involved and includes paths with edges contributed by different digraphs $G_{A_{\ell}}=(\mathcal{V},\mathcal{E}_{\ell})$ occurring on the common vertex set $\mathcal{V}$. To cope with this generalization, we assign the weighted edges $\varepsilon_{A_{\ell}}\in\mathcal{E}_{\ell}$ of each digraph   $G_{A_{\ell}}$ with a unique color, say $c_{\ell}$, and define the colored digraph

[TABLE]

to have the vertex set $\mathcal{V}=\{1,\dots,n\}$ and a set of edges $\bigcup_{t=1}^{m}\mathcal{E}_{\ell_{t}}$ colored $c_{\ell_{t}}$, obtained from $G_{A_{\ell_{t}}}$. This digraph may have multiple edges with different colors, an edge of $G_{A_{\ell_{1}}\cdots A_{\ell_{m}}}$ is denoted by $[\varepsilon_{A_{\ell}}]_{i,j}$.

In this setting, the $(i,j)$-entry of a matrix product $A_{\ell_{1}}\cdots A_{\ell_{m}}$ corresponds to the highest weight of all colored paths $[\varepsilon_{A_{\ell_{1}}}]_{i_{1},j_{1}},\dots,[\varepsilon_{A_{\ell_{m}}}]_{i_{m},j_{m}}$ of length $m$ from $i=i_{1}$ to $j=j_{m}$ in the digraph $G_{A_{\ell_{1}}\cdots A_{\ell_{m}}}$, where each edge $[\varepsilon_{A_{\ell_{t}}}]_{i_{t},j_{t}}$ has color $c_{\ell_{t}}$, $t=1,\dots,m$. Namely, every edge of the path is contributed uniquely by the digraph $G_{A_{\ell}}$, respecting the color ordering determined by the multiplication concatenation $A_{\ell_{1}}\cdots A_{\ell_{m}}$. Working with a colored digraph $G_{A_{\ell_{1}}\cdots A_{\ell_{m}}}$, we always restrict to colored paths, called properly colored paths, that respect the sequence of coloring $c_{\ell_{1}},\dots,c_{\ell_{m}}$. For this reason, we use the awkward notation $G_{A_{\ell_{1}}\cdots A_{\ell_{m}}}$ that records the multiplication concatenation $A_{\ell_{1}}\cdots A_{\ell_{m}}$.

Notation 3.5**.**

*A matrix product $U=A_{\ell_{1}}\cdots A_{\ell_{m}}$ is denoted as ${U^{\dagger}}$ to indicate that $U$ is realized as the word obtained by the concatenation of the symbols $``A_{1}",\dots,``A_{m}"$, accordingly $G_{A_{\ell_{1}}\cdots A_{\ell_{m}}}$ is denoted by $G_{U^{\dagger}}$, while $U=(u_{i,j})$ stands for the actual result of the matrix product. A path $\gamma_{i,j}$ in $G_{U^{\dagger}}$ is denoted by $[\gamma_{U^{\dagger}}]_{i,j}$ to indicate the sequence of its edges’ coloring determined by ${U^{\dagger}}$. *

In this notation, an entry $u_{i,j}$ of $U=A_{\ell_{1}}\cdots A_{\ell_{m}}$ encodes the highest weight over all properly colored paths $[\gamma_{U^{\dagger}}]_{i,j}$, determined by the word ${U^{\dagger}}$, of length $m$ from $i$ to $j$ in the colored digraph $G_{U^{\dagger}}$. Conversely, the word ${U^{\dagger}}$ can be recovered from the coloring sequence of the edges composing any path $\gamma_{i,j}$ of length  $m$ in $G_{U^{\dagger}}$.

3.3. Synoptic matrices

We begin with a certain class of tropical matrices of a special characteristic.

Definition 3.6**.**

A synoptic matrix is a matrix $A=(a_{i,j})$ in ${\operatorname{Mat}}_{n}(\mathbb{T})$ in which

[TABLE]

for all $i=1,\dots,n-1$ and $j=2,\dots,n$.

We denote the set of all synoptic matrices by $\operatorname{Syn}_{n}(\mathbb{T})$. By definition, in any synoptic matrix:

1. (a)

Each row has a nondecreasing order (from left to right); 2. (b)

Each column has a nonincreasing order (from top to bottom).

In particular, the zero matrix $(\mathbb{0})$ is also synoptic,

Lemma 3.7**.**

$\operatorname{Syn}_{n}(\mathbb{T})$* is a subsemiring in ${\operatorname{Mat}}_{n}(\mathbb{T})$.*

Proof.

To see that $\operatorname{Syn}_{n}(\mathbb{T})$ is closed for multiplication, let $C=AB$ and compute $C=(c_{i,j})$ as

[TABLE]

Hence property (3.7) is preserved. The verification that $\operatorname{Syn}_{n}(\mathbb{T})$ is closed for addition is immediate. ∎

3.4. Corner and flat matrices

In the next subsections we develop a theory of special types of tropical matrices, brought here in its full generality, to be used also for future applications.

Notation 3.8**.**

Given a matrix $A$, we denote by $\operatorname{Row}(A)$ and $\operatorname{Col}(A)$ the set of its rows and columns, respectively. We write $A[I,{\underline{\phantom{x}}\;}]$ for the restriction of $A$ to rows $I\subseteq\operatorname{Row}(A)$ and $A[i,{\underline{\phantom{x}}\;}]$ for the row ${\bf r}_{i}\in\operatorname{Row}(A)$. Similarly, $A[{\underline{\phantom{x}}\;},J]$ stands for the restriction of $A$ to columns $J\subseteq\operatorname{Col}(A)$, and $A[{\underline{\phantom{x}}\;},j]$ for the column ${\bf c}_{j}\in\operatorname{Col}(A)$. $A[I,J]$ denotes the restriction of $A$ the rows $I\subseteq\operatorname{Row}(A)$ and to the columns $J\subseteq\operatorname{Col}(A)$. When $A$ is a square matrix we say that $A[I,J]$ is principal submatrix if $J=I$.

We start with structure matrices, provided by the structure map $\widetilde{\pi}:{\operatorname{Mat}}_{n}(\mathbb{T})\longrightarrow{\operatorname{Mat}}_{n}(\mathbb{B})$, cf. (3.5), and consider first the combinatorial shape of matrices. In this setup, we ignore the values of nonzero entries of matrices (these values will show up later) and identify all of them with $1\in\mathbb{B}$.

Definition 3.9**.**

A corner matrix is a matrix whose top-right corner block (possibly empty) contains only nonzero entries and all entries out of this block are $\mathbb{0}$. A matrix $A\in{\operatorname{Mat}}_{n}(\mathbb{T})$ is called $(\mathfrak{p},\mathfrak{q})$-corner if its corner-block $B_{\mathfrak{p},\mathfrak{q}}$ is $A[I_{\mathfrak{p}},J_{\mathfrak{q}}]$, where

[TABLE]

i.e., $a_{i,j}\neq\mathbb{0}$ for all $(i,j)\in I_{\mathfrak{p}}\times J_{\mathfrak{q}}$, otherwise $a_{i,j}=\mathbb{0}$.

Two corner matrices $A$ and $B$ in ${\operatorname{Mat}}_{n}(\mathbb{T})$ are said to be block-similar, written $A\sim_{\operatorname{blk}}B$, if $\widetilde{\pi}(A)=\widetilde{\pi}(B)$, i.e., both are $(\mathfrak{p},\mathfrak{q})$-corner for some $(\mathfrak{p},\mathfrak{q})\in N\times N$. When $\widetilde{\pi}(A)\leq\widetilde{\pi}(B)$, we write $A\subseteq_{\operatorname{blk}}B$, which means that the corner block of $A$ is contained in that of $B$.

The block indexing $(\mathfrak{p},\mathfrak{q})$ indicates the position of the bottom left corner of a block, and thus uniquely determines the corner block. Note that for boolean matrices the block inclusion $\subseteq_{\operatorname{blk}}$ is compatible with the matrix (partial) order (3.3), in the sense that $\widetilde{\pi}(A)\subseteq_{\operatorname{blk}}\widetilde{\pi}(B)$ implies $\widetilde{\pi}(A)\leq\widetilde{\pi}(B)$.

It easy to verify that the set of all corner matrices forms a multiplicative monoid in ${\operatorname{Mat}}_{n}(\mathbb{T})$, which we denote by $\operatorname{Cor}_{n}(\mathbb{T})$.

Remark 3.10**.**

Suppose ${A}_{{t_{u}}}\in\operatorname{Cor}_{n}(\mathbb{T})$, $t_{u}\in N$ with $u=1,2,3$, are $(\mathfrak{p}_{t_{u}},\mathfrak{q}_{t_{u}})$-corner matrices, i.e., matrices whose corner blocks are given by ${A}_{t_{u}}[I_{t_{u}},J_{t_{u}}]$, with $I_{t_{u}}=\{1,\dots\mathfrak{p}_{t_{u}}\}$ and $J_{t_{u}}=\{\mathfrak{q}_{t_{u}},\dots,n\}$. Then, letting $s=t_{1}$ $t=t_{2}$, and $r=t_{3}$, we have the following:

1. (i)

If $\mathfrak{p}_{s}<\mathfrak{p}_{t}$ then $I_{s}\subset I_{t}$, while if $\mathfrak{q}_{s}<\mathfrak{q}_{t}$ then $J_{s}\supset J_{t}$. 2. (ii)

If $\mathfrak{q}_{s}>\mathfrak{p}_{t}$, that is $J_{s}\cap I_{t}=\emptyset$, then $A_{s}A_{t}=(\mathbb{0})$. 3. (iii)

If $\mathfrak{q}_{s}\leq\mathfrak{p}_{t}$ then $A_{s}A_{t}$ is $(\mathfrak{p}_{s},\mathfrak{q}_{t})$-corner, whose corner block is $(A_{s}A_{t})[I_{s},J_{t}]$.

[TABLE]

In particular, for any $(\mathfrak{p}_{t},\mathfrak{q}_{t})$-corner matrix $A_{t}$ with $\mathfrak{q}_{t}\leq\mathfrak{p}_{t}$, $(A_{t})^{m}$ is $(\mathfrak{p}_{t},\mathfrak{q}_{t})$-corner for every $m\in\mathbb{N}$. Otherwise, for $\mathfrak{q}_{t}>\mathfrak{p}_{t}$, we get $(A_{t})^{m}=(\mathbb{0}).$ 4. (iv)

The block inclusion $A_{r}A_{t}\subseteq_{\operatorname{blk}}A_{r}A_{s}$ holds iff $\mathfrak{q}_{t}\geq\mathfrak{q}_{s}$ (i.e., $J_{t}\subseteq J_{s}$), while $A_{r}A_{t}\subseteq_{\operatorname{blk}}A_{s}A_{t}$ iff $\mathfrak{p}_{r}\leq\mathfrak{p}_{s}$ (i.e., $I_{r}\subseteq I_{t}$). In general for arbitrary nonzero products (i.e., $\mathfrak{q}_{t_{i}}\leq\mathfrak{p}_{t_{i+1}}$ and $\mathfrak{q}_{s_{j}}\leq\mathfrak{p}_{s_{j+1}}$) we have

[TABLE]

and in particular

[TABLE]

In this paper we are especially interested in $(\mathfrak{p},\mathfrak{q})$-corner matrices obtained as products of $(\mathfrak{p},\mathfrak{p})$-corner matrices, we call the latter $\mathfrak{p}$-corner matrices, for short, and denote them by ${A}_{(\mathfrak{p})}$. These are a special type of triangular $(\mathfrak{p},\mathfrak{q})$-corner matrices with $\mathfrak{p}\geq\mathfrak{q}$. However, we have the following general observation, obtained by Remark  3.10.(ii).

Remark 3.11**.**

*The monoid $\operatorname{Cor}_{n}(\mathbb{T})$ of all $(\mathfrak{p},\mathfrak{q})$-corner $n\times n$ matrices, ordered by inclusion, forms a partial forward semigroup (Definition 2.2). Restricting to a fixed generating subset of $\mathfrak{p}$-corner matrices with different $\mathfrak{p}=1,\dots,n,$ one obtains a finitely generated forward semigroup. *

So far we have concerned only with the structure of corner matrices, rather than the values of their nonzero entries. Considering suitable values for these entries, block inclusions can be translated to matrix inequalities (3.3). To obtain this view we restrict to corner matrices whose nonzero entries all have a same fixed value $\kappa\neq\mathbb{0}$. We call these matrices $\kappa$-flat corner matrices. For every $\kappa$-flat $\mathfrak{p}$-corner matrix  ${F}_{(\mathfrak{p})}$ we have

[TABLE]

for any $m\in\mathbb{N}$. In particular ${F}_{(\mathfrak{p})}$ is idempotent when $\kappa=\mathbb{1}$.

Lemma 3.12**.**

Let ${F}_{(\mathfrak{p})},{F}_{(\mathfrak{q})},{F}_{(\mathfrak{r})}$ be $\kappa$-flat  corner matrices with $\mathfrak{p}\leq\mathfrak{q}\leq\mathfrak{r}$ then:

1. (i)

${F}_{(\mathfrak{q})}{F}_{(\mathfrak{p})}=(\mathbb{0})$* for $\mathfrak{p}<\mathfrak{q}$;* 2. (ii)

${F}_{(\mathfrak{p})}{F}_{(\mathfrak{q})}$* is $(\mathfrak{p},\mathfrak{q})$-corner for $\mathfrak{p}\leq\mathfrak{q}$; * 3. (iii)

${F}_{(\mathfrak{p})}{F}_{(\mathfrak{q})}>{F}_{(\mathfrak{p})}{F}_{(\mathfrak{r})}$* for $\mathfrak{q}<\mathfrak{r}$;* 4. (iv)

${F}_{(\mathfrak{q})}{F}_{(\mathfrak{r})}>{F}_{(\mathfrak{p})}{F}_{(\mathfrak{r})}$* for $\mathfrak{p}<\mathfrak{q}$.*

Proof.

Since $\mathfrak{p}\leq\mathfrak{q}\leq\mathfrak{r}$ we have

[TABLE]

with strict inclusions when $\mathfrak{p}<\mathfrak{q}<\mathfrak{r}$.

(i) and (ii) are a special case of products of corner matrices, pointed in Remak 3.10 (i) and (iii), respectively.

(iii): By ($*$ ‣ 6.3) and Remark 3.10.(iv) we have ${F}_{(\mathfrak{p})}{F}_{(\mathfrak{q})}\supseteq_{\operatorname{blk}}{F}_{(\mathfrak{p})}{F}_{(\mathfrak{r})}$, where each nonzero entry of ${F}_{(\mathfrak{p})}{F}_{(\mathfrak{q})}$ and ${F}_{(\mathfrak{p})}{F}_{(\mathfrak{r})}$ equals $\kappa^{2}$.

(v): Use a similar argument as in (iii), where now ${F}_{(\mathfrak{q})}{F}_{(\mathfrak{r})}\supseteq_{\operatorname{blk}}{F}_{(\mathfrak{p})}{F}_{(\mathfrak{r})}$. ∎

3.5. Tropical linear representations

We write $\mathbb{T}^{(n)}$ for the Cartesian product $\mathbb{T}\times\dots\times\mathbb{T}$, with $\mathbb{T}$ repeated $n$ times, considered as a $\mathbb{T}$-module over the semiring $\mathbb{T}$ (Definition 1.14), whose operations induced by the operations of $\mathbb{T}$. As in this paper we focus on combinatorial aspects, we identify the associative algebra ${\operatorname{Lin}}(\mathbb{T}^{(n)})$ of all tropical linear operators on $\mathbb{T}^{(n)}$ with ${\operatorname{Mat}}_{n}(\mathbb{T})$.

A finite dimensional tropical (linear) representation of a semigroup $\mathcal{S}$, over $\mathbb{T}^{(n)}$, is a semigroup homomorphism

[TABLE]

A representation $\rho$ is said to be faithful if it is an injective homomorphism. As in classical representation theory one should think of a representation as a tropical linear action of $\mathcal{S}$ on the space $\mathbb{T}^{(n)}$ (since to every $a\in\mathcal{S}$, there is an associated tropical linear operator $\rho(a)$ in ${\operatorname{Lin}}(\mathbb{T}^{(n)})$ that acts on $\mathbb{T}^{(n)}$). Tropically, these representations have an extra digraph meaning (cf. §3.2), associating a semigroup element to a weighted digraph where the semigroup operation is interpreted as an action of one digraph on another diagraph.

Tropical linear representations were introduced in [19], applied there to prove that the bicyclic monoid admits the Adjan identity [1]. These representations are a major method in the present paper, implemented first to a finitely generated forward semigroup  (Definition 2.2).

Theorem 3.13**.**

A finitely generated forward semigroup $\mathcal{F}_{n}=\langle\,{\mathfrak{f}_{\ell}\ {|}\ \ell=1,\dots,n}\,\rangle$ has a tropical linear representation

[TABLE]

determined by generators’ mapping $\mathfrak{f}_{\ell}\longmapsto{F}_{(\ell)}$, where ${F}_{(\ell)}$ are $\ell$-corner matrices in ${\operatorname{TMat}}_{n}(\mathbb{T})$.

Proof.

Associativity is clear, while, by Lemma 3.12.(i), $\rho(\mathfrak{f}_{j}\mathfrak{f}_{i})=\rho(\mathfrak{f}_{j})\rho(\mathfrak{f}_{i})={F}_{(j)}{F}_{(i)}=(\mathbb{0})$ for any $j>i$. ∎

Conversely, a finitely generated matrix semigroup $\mathfrak{F}_{n}=\langle\,{{F}_{(1)},\;{F}_{(2)},\;\dots,\;{F}_{(n)}}\,\rangle\subset{\operatorname{TMat}}_{n}(\mathbb{T})$, where ${F}_{(\ell)}$ are $\kappa$-flat  $\ell$-corner matrices of fixed $\kappa$, is a forward semigroup, cf. Remark 3.11. Its nonzero elements are all singular matrices of rank $1$ (Definition 3.4), and thus $\mathfrak{F}_{n}$ is a singular matrix subsemigroup of ${\operatorname{TMat}}_{n}(\mathbb{T})$, cf. [15].

Corollary 3.14**.**

The map

[TABLE]

determined by generators’ mapping $\mathfrak{f}_{\ell}\longmapsto{F}_{(\ell)}$, is a tropical linear representation of the finitely generated forward semigroup $\mathcal{F}_{n}$.

Note that $\rho=\rho_{\operatorname{fw}}$ is not injective, since by (3.10) we have

[TABLE]

and thus it is not a faithful representation.

4. Troplactic matrix algebra

In this section we utilize the $\kappa$-flat corner matrices to construct a specific troplactic algebra, carrying a digraph meaning, to be utilized later for monoid representations. For mater of generality we use $\kappa$ as a parameter, assumed taking generic values $>\mathbb{1}$, as it also supports a polynomial view to matrix invariants, e.g., traces and permanent.

Our setup consists of the following auxiliary matrices in ${\operatorname{TMat}}_{n}(\mathbb{T})\subset{\operatorname{Mat}}_{n}(\mathbb{T})$:

1. (a)

A collection of $\kappa$-flat $\ell$-corner matrices {F}_{(\ell)}=\big{(}{f}^{(\ell)}_{i,j}\big{)}, $\ell=1,\dots,n$, with fixed $\kappa$, defined by

[TABLE]

These matrices ordered as ${F}_{(1)}<{F}_{(2)}<\cdots<{F}_{(n)}$, and by Lemma 3.12 generate the forward matrix semigroup (Definition 2.2)

[TABLE]

cf. (Remark 3.11). (Note that $\mathfrak{F}_{n}$ is a partially ordered semigroup.) 2. (b)

A layout matrix $E$ that is a triangular idempotent matrix such that $E{F}_{(\ell)}=E{F}_{(\ell)}$ for every $\ell=1,\dots,n$, for simplicity this matrix is taken to be the upper triangular matrix $E=(e_{i,j})$ defined as

[TABLE]

Note that $E$ is both multiplicatively and additively idempotent, i.e, $E=E+E=E^{2}$.

The above matrices are used to define the finitely generated matrix algebra

[TABLE]

whose generators are the triangular matrices ${A}_{(\ell)}:=E\vee{F}_{(\ell)}$, ordered as

[TABLE]

and has the multiplicative identity $E$ and zero $(\mathbb{0})$. We write

[TABLE]

for the multiplicative monoid of $\mathfrak{A}_{n}$. (The whole $\mathfrak{A}_{n}$ is a pointed multiplicative semigroup, with absorbing element $(\mathbb{0})$.) The monoid $\mathfrak{A}^{\times}_{n}$ is a nonsingular partially ordered monoid, whose members all have rank $n$, but it is not a cancellative monoid.

We start with the combinatorial structure of the multiplicative monoid $\mathfrak{A}^{\times}_{n}$, that establishes an important linkage to digraphs. The special structure of the generating matrices ${A}_{(\ell)}$, together with their digraph realization, allows to record lengthes of nondecreasing subsequences of letters (i.e., nondecreasing subwords) by the means of matrix multiplication, leading to the following key lemma.

Key lemma 4.1**.**

Let $L_{m}=[\ell_{1},\ell_{2},\dots,\ell_{m}]$ be a sequence of indexes $\ell_{\xi}\in\{1,\dots,n\}$ with $\xi=1,\dots,m$. Let $U=\prod_{\ell_{\xi}\in L_{m}}{A}_{(\ell_{\xi})}$ be a product of generating matrices ${A}_{(\ell_{\xi})}$ in $\mathfrak{A}^{\times}_{n}$, and write $U=(u_{i,j})$. Then, for $i\leq j$, the $(i,j)$-entry $u_{i,j}$ of the matrix $U$ encodes as the power of $\kappa$ the length of the longest nondecreasing subsequence of $L_{m}$ that involves only terms from the (convex) subsequence $[i:j]\sqsubseteq[1:n]$.

In other words, the $(i,j)$-entry $u_{i,j}$ of the matrix $U$ records the length of longest nondecreasing subword of ${U^{\dagger}}$ restricted to the convex sub-alphabet $``{A}_{(i)}",\dots,``{A}_{(j)}"$, cf. Notation 3.5.

Proof.

We employ the digraph realization of tropical matrices, cf. §3.2. In this unique realization the digraph $G_{{A}_{(\ell)}}$ associated to the matrix ${A}_{(\ell)}$ is an acyclic weighted digraph having the following properties:

1. (a)

Every vertex $1,\dots,n$ is assigned with a loops of weight $\mathbb{1}=0,$ except the vertex $\ell$ whose loop $\rho_{\ell}$ has weight $\omega(\rho_{\ell})=\kappa$. 2. (b)

The only directed edges of $G_{{A}_{(\ell)}}$ are

1. (i)

$\varepsilon_{s,\ell}=(s,\ell)$ for every $s<\ell$; 2. (ii)

$\varepsilon_{\ell,t}=(\ell,t)$ for every $t>\ell$; 3. (iii)

$\varepsilon_{s,t}=(s,t)$ only for $s<\ell<t$;

and all of these matrices have weight $\kappa$.

By these properties, given fixed indices $i\leq j$, we see that for all $s<i\leq j<t$ the digraphs $G_{{A}_{(s)}}$ and $G_{{A}_{(t)}}$ have no directed edges $(p,q)$ for any $i\leq p<q\leq j$. Therefore, when considering paths between pairs of vertices in $[i:j]$ we can ignore the digraphs $G_{{A}_{(s)}}$ and $G_{{A}_{(t)}}$ with $s<i$, $t>j$.

Let ${U^{\dagger}}$ denote the restoring of the multiplication sequence ${A}_{(\ell_{1})}\cdots{A}_{(\ell_{m})}$ as a word (Notation 3.5), and let $G_{U^{\dagger}}$ denote the graph

[TABLE]

while $U$ stands for the matrix product. Recall from §3.2 that any considered path $\gamma_{i,j}$ in $G_{U^{\dagger}}$ always respects the edges’ coloring determined by the sequence $[\ell_{1},\ell_{2},\dots,\ell_{m}]$, $\ell_{\xi}\in\{1,\dots,n\}$, or equivalently by ${U^{\dagger}}$, and is denoted by $[\gamma_{U^{\dagger}}]_{i,j}$ to indicate this coloring.

Taking $i\leq j$, an edge $\varepsilon_{i,j}$ from $i$ to $j$ in $G_{U}$ corresponds to a path $[\gamma_{U^{\dagger}}]_{i,j}$ of highest weight from $i$ to $j$ of length  $m$ in $G_{U^{\dagger}}$. By properties (a)-(b) above, we learn that the possible contribution of the diagraphs $G_{{A}_{(s)}}$ and $G_{{A}_{(t)}}$, $s<i\leq j<s$, to $[\gamma_{U^{\dagger}}]_{i,j}$ could only be loops, all having weight $\mathbb{1}=0$. This means, that we can reduce $[\gamma_{U^{\dagger}}]_{i,j}$ to a sub-path $[\gamma_{U^{\prime\dagger}}]_{i,j}$ (exactly of the same weight, but perhaps shorter) whose edges coloring is determined by the subsequence $L^{\prime}_{m^{\prime}}=[\ell^{\prime}_{1},\dots,\ell^{\prime}_{m^{\prime}}]\sqsubseteq[\ell_{1},\ell_{2},\dots,\ell_{m}]$, $m^{\prime}\leq m$, with $\ell^{\prime}_{\xi}\in\{i,\dots,j\}$. Writing ${U^{\prime\dagger}}$ for the multiplication sequence ${A}_{(\ell^{\prime}_{1})}\cdots{A}_{(\ell^{\prime}_{m^{\prime}})}$, realized a word, then $[\gamma_{U^{\prime\dagger}}]_{i,j}$ is a colored path from $i$ to $j$ of highest weight and length $m^{\prime}$ in $G_{U^{\prime\dagger}}$.

Let ${S}^{\uparrow}=[\ell^{\prime\prime}_{1},\ell^{\prime\prime}_{2},\dots,\ell^{\prime\prime}_{m^{\prime\prime}}]\sqsubseteq[\ell^{\prime}_{1},\ell^{\prime}_{2},\dots,\ell^{\prime}_{m^{\prime}}]$, $\ell^{\prime\prime}_{\xi}\in\{i,\dots,j\}$, $m^{\prime\prime}\leq m^{\prime}$, be a longest nondecreasing subsequence of $L^{\prime}_{m^{\prime}}$. Let

[TABLE]

be the (distinct) elements of $\{i,\dots,j\}$ that take part in ${S}^{\uparrow}$, and let $q_{\xi}$, $\xi=1,\dots,r$, be the number of occurrences of $p_{\xi}$ in ${S}^{\uparrow}$. (These occurrences must be sequential occurrences, as ${S}^{\uparrow}$ is nondecreasing.) Accordingly,

[TABLE]

We denote by ${P^{\dagger}}$ the multiplication sequence ${A}_{(\ell^{\prime\prime}_{1})}\cdots{A}_{(\ell^{\prime\prime}_{m^{\prime\prime}})}$, determined by ${S}^{\uparrow}$ and realized as a word. (Namely ${P^{\dagger}}\subseteq_{\operatorname{wd}}{W^{\dagger}}$.)

Assuming first that $\kappa=1$, we claim that $\omega(\gamma^{\prime}_{i,j})=\operatorname{len}({S}^{\uparrow})$ for $\gamma^{\prime}_{i,j}:=[\gamma_{U^{\prime\dagger}}]_{i,j}$. To prove that $\omega(\gamma^{\prime}_{i,j})\geq\operatorname{len}({S}^{\uparrow})$, consider the (colored) path

[TABLE]

of length $m^{\prime\prime}$ in $G_{P^{\dagger}}$, obtained from ${P^{\dagger}}$ by ($*$ ‣ 6.3). All the edges and loops in $\mu_{i,j}:=[\mu_{P^{\dagger}}]_{i,j}$ have weight $1$, and thus $\omega(\mu_{i,j})=m^{\prime\prime}$. Clearly, $\omega(\gamma^{\prime}_{i,j})\geq\omega(\mu_{i,j})$, since otherwise by plugging in loops of weight [math] we could expand the path $\mu_{i,j}$ to a properly colored path $\widetilde{\mu}_{i,j}$ in $G_{U^{\prime\dagger}}$ with $\omega(\widetilde{\mu}_{i,j})=\omega(\mu_{i,j})$ to get that $\omega(\widetilde{\mu}_{i,j})>\omega(\gamma^{\prime}_{i,j})$, contradicting the weight maximality of the path $\gamma^{\prime}_{i,j}$ in $G_{U^{\prime\dagger}}$.

On the other hand, if $\omega(\gamma^{\prime}_{i,j})>\omega(\mu_{i,j})$, as $\gamma^{\prime}_{i,j}$ is an acyclic path, by excluding all loops of weight  [math] we could extract from $\gamma^{\prime}_{i,j}$ a (properly colored) acyclic sub-path $\gamma^{\prime\prime}_{i,j}$ with $\omega(\gamma^{\prime\prime}_{i,j})=\omega(\gamma^{\prime}_{i,j})$. Then, the coloring of edges in $\gamma^{\prime\prime}_{i,j}$ determines a nondecreasing sequences in $L^{\prime}_{m^{\prime}}$ of length greater than $\omega(\mu_{i,j})=\operatorname{len}({S}^{\uparrow})$, contradicting the length maximality of ${S}^{\uparrow}$. Composing together, we therefore have $\omega(\gamma_{i,j})=\omega(\gamma^{\prime}_{i,j})=\omega(\mu_{i,j})=\operatorname{len}({S}^{\uparrow})$.

To complete the proof for a generic $\kappa$, just observe that $m=1^{m}$ and that the map $\kappa^{m}\longmapsto m$, $m\in\mathbb{N}$, is a bijection. ∎

Key Lemma 4.1 has dealt with the multiplicative (monoid) structure of $\mathfrak{A}^{\times}_{n}$, and now we turn to discuss the semiring structure of $\mathfrak{A}_{n}$.

Theorem 4.2**.**

The finitely generated matrix algebra $\mathfrak{A}_{n}:=\langle\,{{A}_{(1)},\dots,{A}_{(n)}}\,\rangle$ of (4.4) is a troplactic algebra $\mathfrak{plc}_{n}$ (Definition 2.4). Thus, every triplet of it generators

[TABLE]

admit the Knuth relations (KNT):

[TABLE]

and these matrix products are different from $ABC$ and $CBA$.

Proof.

We verify the axioms of troplactic algebra in Definition 2.4.

$\operatorname{PA1}$:

$(\mathfrak{a}=\mathfrak{e}+\mathfrak{a}.)$  By construction, and the additive idempotence of ${\operatorname{Mat}}_{n}(\mathbb{T})$ we have

[TABLE]

$\operatorname{PA2}$:

($\mathfrak{b}\mathfrak{a}=\mathfrak{a}+\mathfrak{b}$ if $\mathfrak{b}>\mathfrak{a}$.)  By Lemma 3.12.(i) we have

[TABLE]

$\operatorname{PA3}$:

($\mathfrak{a}(\mathfrak{b}+\mathfrak{c})=\mathfrak{a}\mathfrak{b}+\mathfrak{c}.$)  By Lemma 3.12.(iii) we have

[TABLE]

$\operatorname{PA4}$:

($(\mathfrak{a}+\mathfrak{b})\mathfrak{c}=\mathfrak{a}+\mathfrak{b}\mathfrak{c}.$)  By Lemma 3.12.(iv) we have

[TABLE]

Use Key Lemma  4.1, to see that the products $U=ABC$ and $V=CBA$ are differ from the products $X=ACB$ and $Y=BAC$, which gives $u_{p,r}=\kappa^{3}$, $v_{p,r}=\kappa$ for the $(p,r)$-entry of $U=(u_{i,j})$, $V=(v_{i,j})$ respectively, while $x_{p,r}=y_{p,r}=\kappa^{2}$ for the $(p,r)$-entry of $X=(x_{i,j})$ and $Y=(y_{i,j})$. ∎

The theorem shows that for three letter words $u\equiv_{\operatorname{clk}}v$ iff $u\equiv_{\operatorname{knu}}v$, cf. (5.2), but in general this relation does not hold for words of an arbitrary length, as seen later in Example 6.8.

Proposition 4.3**.**

Any two generators ${A}_{(k)}$ and ${A}_{(\ell)}$ admit the Frobenius property:

[TABLE]

Proof.

Immediate by Lemma 2.15, since $\mathfrak{A}_{n}$ is a troplactic algebra. ∎

Let $W={A}_{(\ell_{1})}\cdots{A}_{(\ell_{m})}$, $W=(w_{i,j})$, be a matrix product in $\mathfrak{A}^{\times}_{n}$, realized as a word ${W^{\dagger}}$ over the symbols $``{A}_{(\ell_{1})}",\dots,``{A}_{(\ell_{m})}"$. Setting $\kappa=1$, the multiplicative monoid $\mathfrak{A}^{\times}_{n}$ has two main characters:

1. (i)

The additive trace

[TABLE]

that provides the maximum occurrences of a letter in ${W^{\dagger}}$. 2. (ii)

The multiplicative trace

[TABLE]

that gives the length of the matrix word ${W^{\dagger}}$.

Considering $\mathbb{N}_{0}$ as the tropical semiring $(\mathbb{N}_{0},\vee,\cdot\,)$, these characters preserve their type, additive and multiplicative respectively,

[TABLE]

for any $U,V\in\mathfrak{A}^{\times}_{n}$. Note that $\chi^{+}(W)=\chi^{\times}(W)$ iff ${W^{\dagger}}$ consists of a single letter, where $\chi^{+}(W)=\chi^{\times}(W)=0$ iff ${W^{\dagger}}=E$. For $\chi^{+}$ and $\chi^{\times}$ we also have the properties

[TABLE]

and hence

[TABLE]

The following conclusion is then evident.

Corollary 4.4**.**

If ${A}_{(\ell_{1})}\cdots{A}_{(\ell_{m})}={A}_{(\ell^{\prime}_{1})}\cdots{A}_{(\ell^{\prime}_{m^{\prime}})}\neq E,$ $\ell_{t},\ell^{\prime}_{s}\in\{1,\dots,n\},$ then $m=m^{\prime}$.

5. The cloaktic  and the co-cloaktic monoids

We gently coarser the Knuth relations (KNT) to introduce a new monoid structure over the alphabet $\mathcal{A}_{I}=\{a_{\ell}\ {|}\ \ell\in I\}$ whose underlying equivalence is based on the forthcoming function. Given a word $w\in\mathcal{A}_{I}$ and a convex set $J\subseteq_{\operatorname{cx}}I$ (Definition 1.1), we define the set ${\operatorname{sw}}_{J}(w)$ of all subwords $u$ of $w$ with letters in the convex sub-alphabet $\mathcal{A}_{J}\subseteq_{\operatorname{cx}}\mathcal{A}_{I}$, i.e.,

[TABLE]

and specify the subset $\overrightarrow{{\operatorname{sw}}}_{J}(w)$ consisting of all nondecreasing words in ${\operatorname{sw}}_{J}(w).$ The function $\overrightarrow{\operatorname{len}}_{J}(w)$ gives that length of the longest nondecreasing word $u\in\overrightarrow{{\operatorname{sw}}}_{J}(w)$, possibly non-unique, that is

[TABLE]

Relying upon on this function, rather than on recombining the underlying relations of $\equiv_{\operatorname{knu}}$ (e.g., as in the Chinese monoid [5]), we define the following new monoid, slightly coarsening the plactic monoid (Definition   2.1).

Definition 5.1**.**

The cloaktic monoid is the monoid $\mathcal{K}_{I}:=\mathsf{CLK}(\mathcal{A}_{I})$, generated by an ordered set of elements 333For simplicity, we assume a countable set of generators, where the generalization to an arbitrary ordered set of generators is obvious. Moreover, one can also generalize this monoid by considering a partially ordered set of generators, but such theory is more involved. $\mathcal{A}_{I}:=\{a_{\ell}\ {|}\ \ell\in I\}$, subject to the equivalence relation $\equiv_{\operatorname{clk}}$, defined as

[TABLE]

Namely $\mathcal{K}_{I}:=\mathcal{A}_{I}^{*}/_{\equiv_{\operatorname{clk}}}$. When $|I|=n$ is finite, we write $\mathcal{K}_{n}$ for $\mathcal{K}_{I}$ and say that $\mathcal{K}_{n}$ is finitely generated of rank $n$.

We write $\mathsf{CLK}_{I}$ for $\mathsf{CLK}(\mathcal{A}_{I})$ when the alphabet $\mathcal{A}_{I}$ is arbitrary. Henceforth, we always assume that $e<a_{\ell}$ for all $a_{\ell}\in\mathcal{K}_{I}$.

Lemma 5.2**.**

The equivalence relation $\equiv_{\operatorname{clk}}$ is a congruence on the free monoid $\mathcal{A}_{I}^{*}$.

Proof.

Suppose that $u_{1}\equiv_{\operatorname{clk}}v_{1}$ and $u_{2}\equiv_{\operatorname{clk}}v_{2}$, and assume that $u_{1}u_{2}\not\equiv_{\operatorname{clk}}v_{1}v_{2}$. Then, say, $\overrightarrow{\operatorname{len}}_{J}(u_{1}u_{2})>\overrightarrow{\operatorname{len}}_{J}(v_{1}v_{2})$ for some $\mathcal{A}_{J}\subseteq_{\operatorname{cx}}\mathcal{A}_{I}$. Let with $w\subseteq_{\operatorname{wd}}u_{1}u_{2}$ a longest nondecreasing subword of $u_{1}u_{2}$ with letters from $\mathcal{A}_{J}$. As such $w$ decomposes as $w=w_{1}w_{2}$, where $w_{1}\subseteq_{\operatorname{wd}}u_{1}$, $w_{2}\subseteq_{\operatorname{wd}}u_{2}$ are nondecreasing. Accordingly $w_{1}\in\mathcal{A}_{J_{1}}^{*}$, $w_{2}\in\mathcal{A}_{J_{2}}^{*}$ such that $\mathcal{A}_{J_{1}}\cap\mathcal{A}_{J_{2}}\subset\mathcal{A}_{J}$ where

[TABLE]

Since $u_{1}\equiv_{\operatorname{clk}}v_{1}$ and $u_{2}\equiv_{\operatorname{clk}}v_{2}$, there are nondecreasing subwords $w^{\prime}_{1}\subseteq_{\operatorname{wd}}v_{1}$ in $\mathcal{A}_{J_{1}}^{*}$ and $w^{\prime}_{2}\subseteq_{\operatorname{wd}}v_{2}$ in $\mathcal{A}_{J_{2}}^{*}$ satisfying $\operatorname{len}(w^{\prime}_{1})=\operatorname{len}(w_{1})$ and $\operatorname{len}(w^{\prime}_{2})=\operatorname{len}(w_{2})$. Then ($*$ ‣ 6.3) ensures that the concatenation $w^{\prime}_{1}w^{\prime}_{2}$ is a nondecreasing subword of $u_{2}v_{2}$ and therefore has length $\operatorname{len}(w_{1})\operatorname{len}(w_{2})$, and hence $\overrightarrow{\operatorname{len}}_{J}(u_{1}u_{2})=\overrightarrow{\operatorname{len}}_{J}(v_{1}v_{2})$ – a contradiction. ∎

When $u\equiv_{\operatorname{clk}}v$, for $u,v\in\mathcal{A}_{I}^{*}$, we say that the words $u$ and $v$ are cloaktically equivalent. That is $u\equiv_{\operatorname{clk}}v$, if over any convex sub-alphabet $\mathcal{A}_{J}$ of $\mathcal{A}_{I}$ the length of longest nondecreasing subwords in $u$ and $v$ is the same. In particular each letter $a_{\ell}\in\mathcal{A}_{I}$ is cloaktically equivalent only to itself.

Example 5.3**.**

The bicyclic monoid is the monoid $\mathcal{B}:=\mathcal{A}_{2}^{*}/_{\equiv_{\operatorname{bcy}}}$ generated by two ordered elements $a<b$, subject to the relation $\equiv_{\operatorname{bcy}}$ determined by $ab=e$. As well known (e.g. [6]), each element of $w\in\mathcal{B}$ can be canonically written as

[TABLE]

Accordingly, for any $w\in\mathcal{B}$ we have

[TABLE]

Thus, we see that on $\mathcal{A}_{2}^{*}$ the congruence $\equiv_{\operatorname{bcy}}$ implies $\equiv_{\operatorname{knu}}$.

By Definition 5.1 we have the following obvious properties.

Properties 5.4**.**

For two cloaktically equivalent words $u\equiv_{\operatorname{clk}}v$ in $\mathcal{K}_{I}$ the below properties hold.

1. (i)

Each letter $a_{i}\in\mathcal{A}_{I}$ appears in $u$ and in $v$ exactly the same times, i.e., the formal relation $u\equiv_{\operatorname{clk}}v$ is balanced (cf. §1.4). 2. (ii)

The total length of $u$ and $v$ is the same, i.e., $\operatorname{len}(u)=\operatorname{len}(v)$. 3. (iii)

If $u$ is nondecreasing, or nonincreasing, then $u=v$. 4. (iv)

The equivalence $u\equiv_{\operatorname{clk}}v$ does not imply that set theoretically $\overrightarrow{{\operatorname{sw}}}_{J}(u)=\overrightarrow{{\operatorname{sw}}}_{J}(v)$, $J\subseteq_{\operatorname{cx}}I$, nor even that longest words in $\overrightarrow{{\operatorname{sw}}}_{J}(u)$ and $\overrightarrow{{\operatorname{sw}}}_{J}(v)$ are the same.

Recall that $\equiv_{\operatorname{plc}}$ denotes congruence determined by the plactic relations, which are the Knuth relations, and hence $\equiv_{\operatorname{plc}}$ is exactly $\equiv_{\operatorname{knu}}$.

Remark 5.5**.**

The monoid homomorphism $\mathcal{A}_{I}^{*}\longrightarrow\mathcal{K}_{I}$ factors through the plactic monoid (Definition  2.1)

[TABLE]

as will be seen later in Proposition 6.9 and Corollary 6.10. Furthermore, from this factorization, we can conclude that elements $w\in\mathcal{K}_{2}$ of the cloaktic monoid of rank $2$ have a canonical form

[TABLE]

generalizing Example 5.3.

5.1. Linear representations of the cloaktic monoid

The troplactic matrix algebra $\mathfrak{A}_{n}=\langle\,{{A}_{(1)},\dots A_{(n)}}\,\rangle$, as defined in (4.4), is now utilized to introduce a linear representation of the finitely generated cloaktic monoid $\mathcal{K}_{n}=\langle\,{a_{1},\dots,a_{n}}\,\rangle$ of rank $n$ (Definition 5.1).

Theorem 5.6**.**

The map

[TABLE]

defined by mapping of generators, i.e.,

[TABLE]

and sends the empty word $e$ to $E$, is a monoid isomorphism – a faithful linear representation of the cloaktic monoid $\mathcal{K}_{n}$ of rank $n$.

Proof.

Let $L_{m}=[\ell_{1},\ell_{2},\dots,\ell_{m}]$ and $L^{\prime}_{m^{\prime}}=[\ell^{\prime}_{1},\ell^{\prime}_{2},\dots,\ell^{\prime}_{m^{\prime}}]$ be two sequences with $\ell_{t},\ell^{\prime}_{t}\in\{1,\dots,n\}$. Write $u=a_{\ell_{1}}\cdots a_{\ell_{m}}$ and $v=a_{\ell^{\prime}_{1}}\cdots a_{\ell^{\prime}_{m^{\prime}}}$ for the corresponding words in $\mathcal{K}_{n}$ and consider their linear representations

[TABLE]

written as $U=(u_{i,j})$ and $V=(v_{i,j})$.

By Key Lemma 4.1, the $(i,j)$-entry $u_{i,j}$, $i\leq j$, of the matrix $U$ gives the length of the longest nondecreasing subsequence of $L_{m}$ that involves terms only from the convex subsequence $[i:j]\sqsubseteq[1:n]$. The same holds for the $(i,j)$-entry $v_{i,j}$ with respect to $L^{\prime}_{m^{\prime}}$. Taking the generators corresponding to $L_{m}$ and $L^{\prime}_{m^{\prime}}$ in $\mathcal{K}_{n}$, by the definition of the congruence $\equiv_{\operatorname{clk}}$, it implies that $u\equiv_{\operatorname{clk}}v$ iff $U=V$. ∎

Corollary 5.7**.**

The cloaktic monoid $\mathcal{K}_{n}$ satisfies the Knuth relations (KNT).

Proof.

$\mho:\mathcal{K}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$ is isomorphism by Theorem 5.6, where $\mathfrak{A}^{\times}_{n}$ satisfies the Knuth relations (KNT) by Theorem  4.2. ∎

Corollary 5.8**.**

The cloaktic monoid $\mathcal{K}_{n}$ admits all the semigroup identities satisfied by ${\operatorname{TMat}}_{n}(\mathbb{T})$, in particular the semigroup identities (3.6).

Proof.

$\mathcal{K}_{n}$ is faithfully represented by $\mathfrak{A}^{\times}_{n}$ – a submonoid of ${\operatorname{TMat}}_{n}(\mathbb{T})$ – which by Theorem 3.2 satisfies the identities (3.6). ∎

The map $\mho$ does not record explicitly longest convex subwords, but only their lengths. However, by Key Lemma 4.1, we see that each diagonal entry $u_{\ell,\ell}$ of $U=\mho(w)$ records precisely the number of times that the letter $a_{\ell}$ appears in $w$, and thus also the total length of $w$ as the multiplicative trace (4.7). The additive trace (4.6) gives the maximal occurrence of a letter,

Lemma 5.9**.**

Let $T_{n}\subset\mathcal{A}_{n}^{*}$ be the subset of all nondecreasing words over the alphabet $\mathcal{A}_{n}$. Then the restriction

[TABLE]

of the homomorphism (5.3) to $T_{n}$ is a bijective map.

Proof.

A (nondecreasing) word $w\in T_{n}$ is written uniquely as

[TABLE]

Let $U=\mho(w)$, $U=(u_{i,j})$, be the image of $w$ in $\mathfrak{A}^{\times}_{n}$. By Key Lemma 4.1 we have

[TABLE]

and therefore the $q_{k}$’s are uniquely computed in terms of the entries of $U$ as

[TABLE]

(Alternatively, $q_{k}=u_{k,k}$, for every $k=1,\dots,n$.) ∎

Example 5.10**.**

We provide the full details of the tropical linear representation

[TABLE]

of the cloaktic monoid $\mathcal{K}_{3}=\mathsf{CLK}(\mathcal{A}_{3})$ of rank $3$, given by the generators’ map

[TABLE]

where $A={A}_{(1)}$, $B={A}_{(2)}$, $C={A}_{(3)}$, and $\kappa=1$. (Recall that $\mathbb{1}:=0$, and the empty space stand for $\mathbb{0}:=-\infty$.) For these matrix generators we obtain

[TABLE]

where for pairs of generators we have

[TABLE]

[TABLE]

[TABLE]

while

[TABLE]

Finally, by Corollary 5.8, the cloaktic monoid  $\mathcal{K}_{3}$ of rank $3$ satisfies the semigroup identities (3.6), in particular (1.10):

[TABLE]

by letting $x=uv$ and $y=vu$, for any $u,v\in\mathcal{K}_{3}$.

Remark 5.11**.**

Let $\overrightarrow{w}=a_{\ell_{1}}\cdots a_{\ell_{m}}$ be a word in the free monoid $\mathcal{A}^{*}_{n}$, represented in $\mathfrak{A}^{\times}_{n}$ by the matrix product ${A}_{(\ell_{1})}\cdots{A}_{(\ell_{m})}$, via the homomorphism $\mho:\mathcal{A}_{n}^{*}\longrightarrow\mathfrak{A}^{\times},$ cf. (5.3). The reversal $\overleftarrow{w}$ of the word $\overrightarrow{w}$ is represented in $\mathfrak{A}^{\times}_{n}$ by the product

[TABLE]

Indeed, by standard composition of transpositions we have:

[TABLE]

In digraph view, cf. §3.2, the transposition of a matrix $A=(a_{i,j})$ is interpreted as the redirecting of all edges in the associated digraph $G_{A}:=(\mathcal{V},\mathcal{E})$ of $A$. Namely, replacing each edge $\varepsilon_{i,j}\in\mathcal{E}$ by the edge $\varepsilon_{j,i}$ with opposite direction, but with the same weight. Thus, by Key Lemma 4.1, it means that $\mho(\overleftarrow{w})$ gives the lengths of longest nonincreasing subswords in $\overrightarrow{w}$ over the convex sub-alphabets of $\mathcal{A}_{n}$.

Remark 5.12**.**

Two words $u,v\in\mathcal{K}_{n}$ can be cloaktically equivalent, i.e., $u\equiv_{\operatorname{clk}}v$, while their reversals $\overleftarrow{u}$ and $\overleftarrow{v}$ are not cloaktically equivalent, or vise versa. For example, take the words $u=c\;b^{2}c^{2}\;a^{2}b^{2}c$ and $v=c^{2}\;b^{2}c\;a^{2}b^{2}c$ for which $\overleftarrow{u}=cb^{2}a^{2}c^{2}b^{2}c$ and $\overleftarrow{v}=cb^{2}a^{2}cb^{2}c^{2}$. Hence $u\equiv_{\operatorname{clk}}v$ but $\overleftarrow{u}\not\equiv_{\operatorname{clk}}\overleftarrow{v}$.

The example in the remark is a pathological example, and it will have further uses in the paper.

Observation 5.13**.**

Let $u,v\in\mathcal{K}_{n}$ be elements represented in $\mathfrak{A}^{\times}$ by $U=\mho(u)$, $V=\mho(v)$, and consider the matrix sum $W=U+V$. Since by Theorem 5.6 the representation $\mho:\mathcal{K}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$, cf. (5.3), is isomorphism then $w=\mho^{{{\small-1}}}(W)$ is word in $\mathcal{K}_{n}$ that contains $u$ an $v$ as subwords, in which the lengths of longest nondecreasing subwords (5.1) satisfy

[TABLE]

for every convex sub-alphabet $\mathcal{A}_{J}\subseteq_{\operatorname{cx}}\mathcal{A}_{n}.$

From this observation it follows that the additive monoid of $\mathfrak{A}_{n}$ has the structure of a lattice.

5.2. Algorithm and complexity

We utilize our faithful representation of the cloaktic monoid (Theorem 5.6, §5.1) to provide the following efficient algorithm. By “roughly similar length” we mean lengths that are equal up to $\pm 1$.

Algorithm 5.14**.**

Given a word $w\in\mathcal{A}_{n}^{+}$ over a finite alphabet $\mathcal{A}_{n}=\{a_{1},\dots,a_{n}\}$, find the maximal lengthes of its nondecreasing subwords over each convex sub-alphabet $\mathcal{A}_{J}\subseteq_{\operatorname{cx}}\mathcal{A}_{n}$.

1. (1)

Decompose $w$ into two subwords of roughly similar length, repeat this decomposition as long as possible. 2. (2)

Represent each letter $a_{\ell}$ of $w$ by the matrix ${A}_{(\ell)}$ as in (4.4). 3. (3)

Roll back the recursive decomposition of step (1) where in each backward step multiply the resulting matrices obtained from the previous step.

For complexity considerations, to be compatible with the customarily notation, we switch notation and denote the number of letters in the underlining alphabet $\mathcal{A}_{m}$ by $m$; the length of the input (i.e., the length of the input word) is then denoted by $n$. We assume that $\mathcal{A}_{m}$ is a finite alphabet.

Time complxity**.**

Algorithm 5.14 has a linear time complexity $O(n)$.

Proof.

Reading an input word $w\in\mathcal{A}_{m}^{+}$ of length $n$ over a finite alphabet $\mathcal{A}_{m}$, is performed in linear time. Computing the product of two $m\times m$ $\kappa$-flat corner matrices requires a constant time of order   $m^{2}$. As the algorithm is recursive, we have $\log(n)$ matrix multiplications, which in total takes $m^{2}\log(n)$ operations. Putting all together, as $m$ is constant we have $O(\log(n))$ effective operations and reading in $O(n)$, which sum up to $O(n)$. ∎

To the best of our knowledge, no other known algorithm offers such efficient time complexity for this problem.

5.3. The co-cloaktic monoid

In the view of Remark 1.7, we employ the co-mirror of words (Definition 1.6) to introduce the following monoid – a second coarsening of the plactic monoid.

Definition 5.15**.**

The co-cloaktic monoid of rank $n$ is the monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}:={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{CLK}}}(\mathcal{A}_{n})$ generated by a finite ordered set of elements $\mathcal{A}_{n}:=\{a_{1},\dots,a_{n}\}$, subject to the equivalence relation ${\equiv^{\operatorname{co}}_{\operatorname{clk}}}$, defined by (1.3) as

[TABLE]

Namely ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}:=\mathcal{A}_{n}^{*}/_{{\equiv^{\operatorname{co}}_{\operatorname{clk}}}}$.

We write ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{CLK}}}_{n}$ for ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{CLK}}}(\mathcal{A}_{n})$ when the alphabet $\mathcal{A}_{n}$ is arbitrary. As the relation ${\equiv^{\operatorname{co}}_{\operatorname{clk}}}$ is defined in terms of $\equiv_{\operatorname{clk}}$, which is a congruence (Lemma 5.2), respected by the co-mirroring operation (1.4), then it is a congruence. In general the equivalence $\equiv_{\operatorname{clk}}$ on $\mathcal{A}_{n}^{*}$ does not implies the equivalence ${\equiv^{\operatorname{co}}_{\operatorname{clk}}}$, i.e., $u\equiv_{\operatorname{clk}}v$ does not imply $u{\equiv^{\operatorname{co}}_{\operatorname{clk}}}v$ on $\mathcal{A}_{n}^{*}$, or vise versa. This is shown below in Example 7.15, after developing additional methods.

Properties 5.16**.**

For any two words $u{\equiv^{\operatorname{co}}_{\operatorname{clk}}}v$ in ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ we have the following properties:

1. (i)

Every letter $a_{i}\in\mathcal{A}_{n}$ appears in $u$ and in $v$ exactly the same times, i.e., the formal relation $u{\equiv^{\operatorname{co}}_{\operatorname{clk}}}v$ is balanced (cf. §1.4). 2. (ii)

The length of $u$ and $v$ is the same.

As the co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ arises from the equivalence of the cloaktic monoid $\mathcal{K}_{n}$, we manipulate the linear representation of $\mathcal{K}_{n}$, cf. §5.1, to construct a representation of the (finitely generated) co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$. That is, ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}=\langle\,{a_{1},\dots,a_{n}}\,\rangle$ of rank $n$ is represented by the monoid homomorphism (determined by generators’ mapping)

[TABLE]

where $\mathfrak{A}_{n}:=\langle\,{{A}_{(1)},\dots,{A}_{(n)}}\,\rangle$ is the troplactic matrix algebra defined in (4.4). Since $\mathfrak{A}_{n}$ is a troplacitc algebra (Theorem 4.2), then Corollary 2.13 implies the explicitly additive decomposition

[TABLE]

Then ${M}_{(\ell)}$ can be written as

[TABLE]

where ${\widecheck{A}}_{(\ell)}$ is the $n\times n$ matrix of the form

[TABLE]

whose $(\ell^{\prime},\ell^{\prime})$-entry, $\ell^{\prime}=n-\ell+1$, is $\kappa^{{{\small-1}}}$ and its other nonzero entries are all $\mathbb{1}$.

Letting ${(\mathfrak{A}^{\times}_{n})}^{[1]}:=\langle\,{{M}_{(\ell)},\dots,{M}_{(\ell)}}\,\rangle$ be the matrix submonoid of $\mathfrak{A}^{\times}_{n}$ generated by the matrices ${M}_{(\ell)},\dots,{M}_{(\ell)}$ in (5.5) and with identity $E$, from (5.4) we draw the surjective homomorphism

[TABLE]

by restricting the image of $\Omega_{\kappa}$ to ${(\mathfrak{A}^{\times}_{n})}^{[1]}$.

Remark 5.17**.**

Using the endomorphism $\partial_{1}:\mathcal{A}^{*}_{n}\;{\relbar\joinrel\twoheadrightarrow}\;{(\mathcal{A}^{*}_{n})}^{[1]}$ in Remark 1.7, the co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}:=\mathcal{A}_{n}^{*}/_{{\equiv^{\operatorname{co}}_{\operatorname{clk}}}}$ can be defined equivalently as $u{\equiv^{\operatorname{co}}_{\operatorname{clk}}}v\Leftrightarrow\partial_{1}(v)\equiv_{\operatorname{clk}}\partial_{1}(v)$, and thus ${(\mathcal{A}^{*}_{n})}^{[1]}/_{\equiv_{\operatorname{clk}}}$ can be realized as a monoid isomorphic to a submonoid of ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$. (Note that ${(\mathcal{A}^{*}_{n})}^{[1]}/_{\equiv_{\operatorname{clk}}}$ and ${\mathcal{K}_{n}}^{[1]}$ are not necessarily equal as monoids.) On the other hand, \mho:\mathcal{K}_{n}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}\mathfrak{A}^{\times}_{n}, $a_{\ell}\mapsto{A}_{(\ell)}$, is an isomorphism by Theorem 5.6, and its composition with the surjection $\partial_{1}$ shows that $\Omega_{\kappa}:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$, cf. (5.4), is an injective homomorphism. Hence, the map $\Omega_{\kappa}^{[1]}:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\longrightarrow{(\mathfrak{A}^{\times}_{n})}^{[1]}$ is an isomorphism. The following diagram summarizes these homomorphisms

[TABLE]

Yet, we need to complete the lower part of the diagram – the object ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$.

Note also that by Remark 5.5 and Diagram (5.8), the monoid homomorphism $\mathcal{A}_{n}^{*}\longrightarrow{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ factors through the plactic monoid

[TABLE]

We will return to this factorization later.

Despite (5.4) provides a faithful representation of ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ in terms of monoids, i.e., as a matrix submonoid of $\mathfrak{A}^{\times}_{n}$, the images $\Omega(a_{\ell})$ of the generators $a_{\ell}$ of ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ do not generate the image $\Omega({{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n})$ as a tropolactic matrix algebra. To achieve this attribute we define the matrix algebra

[TABLE]

generated by the matrices ${\widecheck{A}}_{(1)},\dots,{\widecheck{A}}_{(n)}$ introduced in (5.6), which we call the co-algebra of $\mathfrak{A}_{n}$. The multiplicative submonoid ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ of ${{}^{\diamond}\joinrel{\mathfrak{A}}}_{n}$ is called the co-monoid of $\mathfrak{A}^{\times}_{n}$, for which we obtain the monoid homomorphism

[TABLE]

i.e., $\Omega(a_{\ell})=\kappa^{{{\small-1}}}\Omega_{\kappa}(a_{\ell})$ for every $a_{\ell}\in\mathcal{A}_{n}$. Accordingly we see that

[TABLE]

is a monoid isomorphism, and the composition

[TABLE]

is a surjective homomorphism.

Applying the negation map (3.4) in Remark 3.1,

[TABLE]

due to the special structure of ${\widecheck{A}}_{(\ell)}$, it only replaces the entry $\widecheck{a}_{\ell^{\prime},\ell^{\prime}}=\kappa^{{{\small-1}}}$ by $\widehat{a}_{\ell^{\prime},\ell^{\prime}}:=\kappa$, where $\ell^{\prime}=n-\ell+1$). The images matrices ${\widehat{A}}_{(\ell)}$ are then considered as matrices in ${\operatorname{Mat}}_{n}(\mathbb{T}_{\wedge})$, whose multiplication are now induced by the semiring operations $+$ and $\wedge$. Nevertheless, as in the sequel we want be consistent with the multiplicative operation of the monoid $\mathfrak{A}^{\times}_{n}$, we stick with the matrices ${\widecheck{A}}_{(\ell)}$ for which the matrix multiplications are taken with respect to semiring operations $+$ and $\vee$ (jointly with negation), while the additive operation $A\wedge B$ can be computed dually as $-(-A\vee-B)$.

Example 5.18**.**

The co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{3}$ of rank 3 is linearly represented by the homomorphism $\Omega:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{3}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ determined by the generators’ mapping

[TABLE]

(Recall that $\mathbb{1}:=0$, and the empty space stand for $\mathbb{0}:=-\infty$.) For these generators we have the products

[TABLE]

Note that ${\widecheck{A}}_{(1)}{\widecheck{A}}_{(2)}={\widecheck{A}}_{(1)}\wedge{\widecheck{A}}_{(2)}$, ${\widecheck{A}}_{(3)}{\widecheck{A}}_{(1)}={\widecheck{A}}_{(3)}\wedge{\widecheck{A}}_{(1)}$, but ${\widecheck{A}}_{(2)}{\widecheck{A}}_{(1)}\neq{\widecheck{A}}_{(2)}\wedge{\widecheck{A}}_{(1)}$

Considering the digraph view of the matrix $W=\Omega(w)$ assigned to a word $w\in\mathcal{A}_{n}^{*}$, cf. §3.2, a nonzero entry $w_{i,j}$ of $W$ corresponds to a path $[\gamma_{W^{\dagger}}]_{i,j}$ of maximal (nonpositive) weight from $i$ to $j$ of length $\operatorname{len}(w)$ in $G_{W^{\dagger}}$ (Notation 3.5). In terms of words, it means the length of the complement of subword of $w$ over $\{a_{i},\dots,a_{j}\}$ of maximal length where each letter has (negative) length $\kappa^{{{\small-1}}}$.

Theorem 5.19**.**

The matrix co-algebra ${{}^{\diamond}\joinrel{\mathfrak{A}}}_{n}:=\langle\,{{\widecheck{A}}_{(1)},\dots,{\widecheck{A}}_{(n)}}\,\rangle$ is a dual troplactic algebra $\mathfrak{plc}^{\wedge}_{n}$ (Definition 2.18), and thus every triplet of matrices

[TABLE]

*admit the Knuth relations (KNT). *

Proof.

We verify the axioms of Definition 2.18 by a straightforward technical computation. By construction

[TABLE]

are the only $\kappa^{{{\small-1}}}$ entries in $\widecheck{A}$, $\widecheck{B}$, and $\widecheck{C}$, with $p^{\prime}>q^{\prime}>r^{\prime}$. All other nonzero entries are $\mathbb{1}$.

1. (a)

The product $\widecheck{A}\widecheck{B}$ has exactly two $\kappa^{{{\small-1}}}$ entries, the $(q^{\prime},q^{\prime})$-entry and the $(p^{\prime},p^{\prime})$-entry, where all other nonzero entries are $\mathbb{1}$. 2. (b)

The product $\widecheck{B}\widecheck{A}$ has at least two $\kappa^{{{\small-1}}}$ entries, the $(q^{\prime},q^{\prime})$-entry and the $(p^{\prime},p^{\prime})$-entry, all other nonzero entries are $\mathbb{1}$. In general, these are the only $\kappa^{{{\small-1}}}$ entries, except in the case that $p^{\prime}=q^{\prime}+1$, in which also the $(q^{\prime},p^{\prime})$-entry has value $\kappa^{{{\small-1}}}$. (See e.g. Example 5.18.)

The other two letter products involving $\widecheck{C}$ have the analogues properties.

$\operatorname{PA1}^{\prime}$:

($\mathfrak{a}=\mathfrak{e}\,\widehat{+}\,\mathfrak{a}$.)  Immediate, by comparison of entries.

$\operatorname{PA2}^{\prime}$:

($\mathfrak{a}\mathfrak{b}=\mathfrak{a}\,\widehat{+}\,\mathfrak{b}$ when $\mathfrak{b}>\mathfrak{a}$.)  A direct implication of property (a), since $p^{\prime}>q^{\prime}$, which is also the structure of the matrix $\widecheck{A}\wedge\widecheck{B}$.

$\operatorname{PA3}^{\prime}$:

($(\mathfrak{b}\,\widehat{+}\,\mathfrak{c})\mathfrak{a}=\mathfrak{b}\mathfrak{a}\,\widehat{+}\,\mathfrak{c}$.)  By property (b) the $\kappa^{{{\small-1}}}$ entries of $\widecheck{B}\widecheck{A}\wedge\widecheck{C}\widecheck{A}$ are $(r^{\prime},r^{\prime})$, $(q^{\prime},q^{\prime})$, and $(p^{\prime},p^{\prime})$, and possibly $(r^{\prime},p^{\prime})$ or $(q^{\prime},p^{\prime})$. If the $(q^{\prime},p^{\prime})$-entry in $\widecheck{B}\widecheck{A}$ is $\kappa^{{{\small-1}}}$ then it is also $\kappa^{{{\small-1}}}$ in $\widecheck{B}\widecheck{A}\wedge\widecheck{C}\widecheck{A}$, independently on $\widecheck{C}\widecheck{A}$. If the $(r^{\prime},p^{\prime})$-entry is $\kappa^{{{\small-1}}}$, then the $(q^{\prime},p^{\prime})$-entry is also $\kappa^{{{\small-1}}}$ and hence $\widecheck{B}=\widecheck{C}$. Thus, in all cases, $(\widecheck{B}\wedge\widecheck{C})\widecheck{A}=\widecheck{B}\widecheck{A}\wedge\widecheck{C}$.

$\operatorname{PA4}^{\prime}$:

($\mathfrak{c}(\mathfrak{b}\,\widehat{+}\,\mathfrak{a})=\mathfrak{a}\,\widehat{+}\,\mathfrak{c}\mathfrak{b}$.)  By property (b) the $\kappa^{{{\small-1}}}$ entries of $\widecheck{C}\widecheck{B}\wedge\widecheck{C}\widecheck{A}$ are $(r^{\prime},r^{\prime})$, $(q^{\prime},q^{\prime})$, and $(p^{\prime},p^{\prime})$, and possibly $(r^{\prime},p^{\prime})$ or $(r^{\prime},q^{\prime})$. If the $(r^{\prime},q^{\prime})$-entry in $\widecheck{C}\widecheck{B}$ is $\kappa^{{{\small-1}}}$ then it is also $\kappa^{{{\small-1}}}$ in $\widecheck{C}\widecheck{B}\wedge\widecheck{C}\widecheck{A}$, independently on $\widecheck{C}\widecheck{A}$. If the $(r^{\prime},p^{\prime})$-entry is $\kappa^{{{\small-1}}}$, then the $(r^{\prime},q^{\prime})$-entry is also $\kappa^{{{\small-1}}}$, which implies $\widecheck{A}=\widecheck{B}$. Thus, in all cases, $\widecheck{C}(\widecheck{B}\wedge\widecheck{A})=\widecheck{A}\wedge\widecheck{C}\widecheck{B}$.

The proof is then completed by Theorem 2.20. ∎

Theorem 5.20**.**

The map

[TABLE]

defined by generators’ mapping, i.e.,

[TABLE]

is a monoid isomorphism – a faithful linear representation of the co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ of rank $n$.

Proof.

The monoid homomorphism $\Omega_{\kappa}:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$ is injective by Remark 5.17, providing the isomorphism \Omega_{\kappa}^{[1]}:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}{(\mathfrak{A}^{\times}_{n})}^{[1]}, which translates to $\Omega:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ by the isomorphism (5.11). ∎

By this theorem we see that $u{\equiv^{\operatorname{co}}_{\operatorname{clk}}}v$ in ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ implies that $\Omega(u)\equiv_{\operatorname{knu}}\Omega(v)$ in ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$, but the converse does not hold as seen later in Example 7.14.

Corollary 5.21**.**

The co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ satisfies the Knuth relations (KNT).

Proof.

$\Omega:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ is an isomorphism, by Theorem 5.20, where ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ by itself admits the Knuth relations by Theorem 5.19. ∎

Corollary 5.22**.**

The co-cloaktic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ admits all the semigroup identities satisfied by ${\operatorname{TMat}}_{n}(\mathbb{T})$, in particular the semigroup identities (3.6).

Proof.

${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ is faithfully represented by ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ – a submonoid of ${\operatorname{TMat}}_{n}(\mathbb{T})$ – which by Theorem 3.2 satisfies the identity (3.6). ∎

6. Configuration tableaux

Towards the goal of introducing linear representations of the plactic monoid $\mathcal{P}_{I}$ (Definition 2.1), we need to establish its precise linkage to the cloaktic monoid $\mathcal{K}_{n}$. To do so we utilize the known bijective correspondence of $\mathcal{P}_{I}$ to Young tableaux [43], therefore by this study we also obtain a useful algebraic description of the latter combinatorial objects. To simplify indexing notations we denote arbitrary letters in $\mathcal{A}_{I}$ by $x,y,$ and $z$.

6.1. Young tableaux

A Young diagram (also called Ferrers diagram) is a finite collection of cells, arranged in left-justified rows, such that the number of cells at each row (i.e., the row’s length) is less or equal than its predecessor. In this paper we use the France convention in which the longest row is the bottom row, and rows are enumerated from bottom to top [30]. Over this setting, we define three types of (proper) steps from a cell to its neighbor:

[TABLE]

1. (a)

a vertical step – a move from a cell to its neighbor below; 2. (b)

a slant step – a move from a cell to its its bottom right neighbor; 3. (c)

a horizontal step – a move from a cell to its right neighbor.

These steps are combined to walks and covers in a Young diagram:

[TABLE]

1. (A)

a falling walk is a continues composition of sequential vertical steps or slant steps; 2. (B)

a descending walk is a continues composition of sequential horizontal steps or slant steps; 3. (C)

a horizontal cover is a collection of cells, one from each column, taken from a nonincreasing subset of rows.

(Note that all of these are “non-left oriented”.) A proper walk is either a falling walk or a descending walk. This terminology is consistent with the content of the Young diagram, as described next. In what follows, all walks and covers are assumed to be proper.

A Young tableau is a Young diagram whose cells are filled by symbols – a single letter in each cell – taken from a given alphabet  $\mathcal{A}_{I}$, usually required to be finite and totally ordered.

Definition 6.1**.**

The track $\operatorname{trk}(\gamma)$ of a (proper) walk $\gamma$ in a Young tableau $\mathfrak{T}$ is the word defined by concatenating the symbols appearing in cells along $\gamma$. The length $\operatorname{len}(\gamma)$ of a walk $\gamma$ is the number of cells along $\gamma$.

When the content of cells in $\mathfrak{T}$ are numbers, the weight of $\gamma$, denoted $w(\gamma)$, is defined as the sum of cells’ values along $\gamma$.

While a walk between two cells needs not be unique, except when the walk is along the bottom row or the left column, all the tracks between pair of cells always have the same length; different tracks may have a same weight.

Young tableaux are combinatorial structure, used intensively in group theory and representation theory  [8]. They provide a useful machinery that led to an elegant algorithmic solution (Schensted 1961) for the following problem:

Problem**.**

Given a (finite) word $w\in\mathcal{A}_{I}^{+}$ on the (totally) ordered alphabet $\mathcal{A}_{I}$, find the length of its longest nondecreasing subwords of $w$.

The central advantage of Schensted’s method is that it avoids a precise identification of a longest nondecreasing subword, and presents a useful linkage between semigroups and Young tableaux, as summarized below.

A nondecreasing word $w\in\mathcal{A}_{I}^{+}$ is called a row, while a (strictly) decreasing word is called a column. We say the row $u=x_{1}\cdots x_{s}$ dominates the row $v=y_{1}\cdots y_{t}$, written $u\rhd v$, if $s\leq t$ and $x_{i}>y_{i}$ for all $i=1,\dots,s$. Any word $w\in\mathcal{A}_{I}^{+}$ has a unique factorization $w=\mathfrak{r}_{p}\cdots\mathfrak{r}_{1}$ as a concatenation of rows $\mathfrak{r}_{i}$ of maximal length.

Definition 6.2**.**

A semi-standard tableau $\mathfrak{T}$ is a tableau that corresponds (by rows) to a word $w\in\mathcal{A}_{I}^{+}$ such that $\mathfrak{r}_{p}\rhd\mathfrak{r}_{p-1}\rhd\cdots\rhd\mathfrak{r}_{1}$, where $\mathfrak{r}_{1}$ is the bottom row and $\mathfrak{r}_{p}$ is the top row of $\mathfrak{T}$. The set of all semi-standard tableaux is denoted by $\mathsf{Tab}(\mathcal{A}_{I})$, written $\mathsf{Tab}_{I}$ for short.

We formally adjoin the empty tableau, denoted $\mathfrak{T}_{0}$, considered as a semi-standard tableau.

Namely, a tableau is semi-standard if the entries in each row are nondecreasing and the entries in each column are increasing. In the sequel we assume that $I$ is finite and write $\mathsf{Tab}_{n}$ for $\mathsf{Tab}_{I}$. A tableau is called standard if the entries in each row and in each column are increasing, which implies that each letter may appear only once. The subset of all standard tableaux over a finite alphabet $\mathcal{A}_{n}$ is denoted by $\mathsf{STab}_{n}.$

We denote the columns of a given tableau $\mathfrak{T}$ by $\mathfrak{c}_{1},\dots,\mathfrak{c}_{q}$, enumerated from left to right. Reading the columns from bottom to top, each column $\mathfrak{c}_{j+1}$ is increasing but not necessarily dominates the column $\mathfrak{c}_{j}$ unless, $\mathfrak{T}$ is standard. In latter case, we define the transpose tableau $\mathfrak{T}^{\operatorname{t}}$ of $\mathfrak{T}$ by writing the columns as rows, i.e., $\mathfrak{c}_{q}\cdots\mathfrak{c}_{1}$, for which $(\mathfrak{T}^{\operatorname{t}})^{\operatorname{t}}=\mathfrak{T}$.

The nonincreasing sequence $\lambda:=(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{k})$ of the rows’ length $\lambda_{i}$ of a tableau $\mathfrak{T}$ defines the shape of $\mathfrak{T}$ and introduces a partition of the integer $|\lambda|=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{k}$, which in its turn determines the graphical representation of the corresponding Young diagram. (The shape of the empty tableau is set to be [math].)

Schensted’s algorithm associates each word $w\in\mathcal{A}_{n}^{+}$ with a tableau $\mathfrak{T}={\operatorname{tab}}(w)$, via the map 444In the literature, the map ${\operatorname{tab}}$ is also denoted by  $P$.

[TABLE]

defined inductively as ${\operatorname{tab}}(wx)={\operatorname{tab}}({\operatorname{tab}}(w)x)$ for an arbitrary $x\in\mathcal{A}_{n}$. It is recursively defined in terms of tableaux as

[TABLE]

where $\mathfrak{r}_{p}\cdots\mathfrak{r}_{1}$ are the row decomposition of $\mathfrak{T}$, $y$ is the left most letter of $\mathfrak{r}^{\prime}_{1}$ which is strictly grater than  $x$, and $\mathfrak{r}^{\prime}_{1}$ is obtained from $\mathfrak{r}_{1}$ by replacing $y$ with $x$.

In a more lingual terms, the map ${\operatorname{tab}}$ can be described in algorithmic terms.

Algorithm 6.3** (Bumping Algorithm).**

Given a word $w=x_{1}\cdots x_{m}$, $x_{i}\in\mathcal{A}_{n}$, start with $\{x_{1}\}$, which is a Young tableau. Suppose $x_{1},\dots,x_{k}$ have already been inserted, and $\mathfrak{T}$ is the current tableau. To insert $x_{k+1}$, start with the first row of $\mathfrak{T}$ and search for the first letter which is greater than $x_{k+1}$. If there is no such element, append $x_{k+1}$ to the end of first row. If there is such an element (say, $x_{j}$), exchange $x_{j}$ by $x_{k+1}$, and proceed inductively to insert $x_{j}$ to the next row.

For three letters $x<y<z$ in $\mathcal{A}_{n}^{+}$, the algorithm produces the familiar tableaux:

[TABLE]

For two distinct letters $x<y$ it gives:

[TABLE]

One immediately sees that the upper tableaux are just realization of the Knuth relations (KNT).

Theorem 6.4** (Schensted 1961).**

The length of a longest nondecreasing (reps. decreasing) subword of $w$ is equal to the length of the bottom row $\mathfrak{r}_{1}$ (resp. height of the first column $\mathfrak{c}_{1}$) of ${\operatorname{tab}}(w)$.

This tableau form introduces on $\mathcal{A}_{n}^{+}$ the equivalence relation

[TABLE]

We call ${\operatorname{tab}}(w)$ the (canonical) tableau form of the word $w\in\mathcal{A}_{n}^{+}$, and write

[TABLE]

Conversely, every tableau $\mathfrak{T}\in\mathsf{Tab}_{n}$ can be identified with $\mathfrak{T}_{w}={\operatorname{tab}}(w)$ for some (not necessarily unique) word $w\in\mathcal{A}^{+}_{n}$. Therefore, we may think of $\mathfrak{T}_{w}$ also as a word having a canonical form in $\mathcal{A}_{n}^{+}$. The extension of ${\operatorname{tab}}:\mathcal{A}_{n}^{+}\longrightarrow\mathsf{Tab}_{n}$ to

[TABLE]

is defined naturally by sending $e\longmapsto\mathfrak{T}_{0}$.

Theorem 6.5** (Knuth 1970).**

The equivalence relation $\equiv_{\operatorname{tab}}$ coincides with the plactic congruence $\equiv_{\operatorname{plc}}$ which is the Knuth congruence $\equiv_{\operatorname{knu}}$. In particular, each plactic class contains exactly one tableau.

In our notation, the theorem reads as

[TABLE]

and hence we may alternate between these notations. Therefore $\mathsf{Tab}_{n}$ can be realized as an algebraic structure, equipped with a multiplicative operation.

Remark 6.6**.**

Set theoretically, $\mathsf{Tab}_{n}$ bijectively corresponds to the plactic monoid $\mathcal{P}_{n}$, thus it can be considered as a monoid whose operation is now induced from Algorithm 6.3, or equivalently from  (6.1). That is, the product of tableaux $\mathfrak{T}_{u}\cdot\mathfrak{T}_{v}$, with say $\mathfrak{T}_{v}=y_{1}\cdots y_{m}$, is defined by the sequential insertion (by reverse row ordering) of the letters $y_{1},\dots,y_{m}$ of $\mathfrak{T}_{v}$ to $\mathfrak{T}_{u}$.

In order to extract numerical invariants from the combinatorial structure of semi-standard tableaux we define the following functions.

1. (a)

The function

[TABLE]

counts the occurrences of the letter $a_{\ell}\in\mathcal{A}_{n}^{+}$ in the $i$’th row of a tableau $\mathfrak{T}\in\mathsf{Tab}_{n}$. 2. (b)

The function

[TABLE]

assigns a tableau with length of its longest nondecreasing subword that involves only letters $\alpha_{\ell}\in\mathcal{B}_{J}$ from the sub-alphabet $\mathcal{B}_{J}\sqsubseteq\mathcal{A}_{n}$.

Our main interested is in convex sub-alphabet $\mathcal{A}_{J}\subseteq_{\operatorname{cx}}\mathcal{A}_{n}$, i.e., $J=\{i,i+1,\dots,j\}$, with $i\leq j$, which we denote as $[a_{i}:a_{j}]$ or $[i:j]$, for short.

When $i=j=\ell$, the function $\natural_{[\ell:\ell]}(\mathfrak{T})$ gives the number of occurrences of the letter $a_{\ell}$ in the tableau  $\mathfrak{T}$, and thus

[TABLE]

We write $\mathfrak{T}|_{[i:j]}$ for the restriction of the tableau $\mathfrak{T}$ to rows $i,\dots,j$.

Remark 6.7**.**

The equivalence relation $\equiv_{\operatorname{clk}}$ on tableaux $\mathfrak{T}_{u}$ and $\mathfrak{T}_{v}$ in $\mathsf{Tab}_{n}$ (cf. (5.2)) reads in terms of  (6.3) as

[TABLE]

(Note that if $\mathfrak{T}_{u}\equiv_{\operatorname{clk}}\mathfrak{T}_{v}$, then $\mathfrak{T}_{u}$ and $\mathfrak{T}_{v}$ must contains exactly the same number of each letter.)

We start with our running example (appeared also in Remark 5.12) that points out special pathologies along our exposition.

Example 6.8**.**

The words $u=c\ b^{2}c^{2}\ a^{2}b^{2}c$ and $v=c^{2}\ b^{2}c\ a^{2}b^{2}c$ have the following tableau realizations:

[TABLE]

and

[TABLE]

It easy to check that $u\equiv_{\operatorname{clk}}v$, where $\natural_{[b:c]}(\mathfrak{T}_{u})=\natural_{[b:c]}(\mathfrak{T}_{v})=5$, but $u\not\equiv_{\operatorname{tab}}v$.

Hence, from this example, we learn that $\equiv_{\operatorname{clk}}$ does not imply $\equiv_{\operatorname{tab}}$ (or equivalently $\equiv_{\operatorname{knu}}$), but the converse holds.

Proposition 6.9**.**

If $u\equiv_{\operatorname{knu}}v$, or equivalently $u\equiv_{\operatorname{tab}}v$, then $u\equiv_{\operatorname{clk}}v$, for any $u,v\in\mathcal{A}^{*}_{n}$. Thus $u\equiv_{\operatorname{clk}}v$ iff $\mathfrak{T}_{u}\equiv_{\operatorname{clk}}\mathfrak{T}_{v}$.

Proof.

By Theorem 6.5, $u\equiv_{\operatorname{knu}}v$ iff $u\equiv_{\operatorname{tab}}v$ iff $\mathfrak{T}_{u}=\mathfrak{T}_{v}$, then it follows from Remark 6.7 that $u\equiv_{\operatorname{clk}}v.$ ∎

Corollary 6.10**.**

The map

[TABLE]

is a surjective homomorphism.

Proof.

Both $\mathcal{P}_{n}$ and $\mathcal{K}_{n}$ admit the Knuth relations (KNT), cf. Corollary 5.7, where $\equiv_{\operatorname{knu}}$ implies $\equiv_{\operatorname{clk}}$ by Proposition 6.9. Thus $\eth$ is a well defined monoid homomorphism. Surjectivity is clear. ∎

Accordingly the diagram

[TABLE]

commutes.

6.2. Configuration tableaux

We introduce a new class of tableaux, carrying nonnegative integers. These tableaux record lengths of tracks in semi-standard tableaux as weights, and are used as axillary tableaux, playing a major role in this paper. The cells’ indexing in these tableaux are arranged in a way that each diagonal corresponds to a letter, as explained below. As will be seen later, this indexing system makes sense in the passage to matrices which establish linear representations of Young tableaux.

Definition 6.11**.**

An $n$-configuration tableau $\mathfrak{C}=(\lambda_{i,j})$ is an isosceles tableau with $n$ rows and $n$ columns of fixed shape $(n>n-1>\cdots>1)$ whose cells are indexed as

[TABLE]

the rows are enumerated from bottom to top and columns from left to right. The index $i$ of $\lambda_{i,j}$ stands for $i$’th row, while $j$ refers to the $j$’th diagonal (enumerated for bottom left).

The content of cells are nonnegative integers $\lambda_{i,j}$ satisfying the following configuration laws for all $i\leq j\leq n$

[TABLE]

The null configuration tableau, denoted $\mathfrak{C}_{0}$, is the $n$-configuration tableau with $\lambda_{i,j}=0$ for all $i,j$. We denote the set of all $n$-configuration tableaux by $\mathsf{CTab}_{n}$.

The $\ell$’th diagonal of $\mathfrak{C}$, for fixed $\ell$, is the the collection of cells $\lambda_{t,\ell}$, $t=1,\dots,\ell$. We define

[TABLE]

which we call the $\ell$’th trace of $\mathfrak{C}$.

An $n$-configuration tableau is called standard if $\tau_{\ell}=1$ for every $\ell=1,\dots,n.$ The set of all standard $n$-configuration tableaux is denoted by $\mathsf{SCTab}_{n}$.

Literally, the configuration laws (6.4) assert that the sum of cells’ values of a left adjacent subrow of $\mathfrak{r}_{i}$ is greater or equal than the sum of cells’ values of any left adjacent subrow of the same length placed above $\mathfrak{r}_{i}$.

Note that configuration tableaux by themselves need not be semi-standard as they may contain [math]-valued cells. Nevertheless, as proved later, they bijectively correspond to semi-standard tableaux. The fact that all $n$-configuration tableaux have the same fixed shape, together with a canonical indexing system of cells, permits uniform formal references, and provides a suitable notation for walks between cells.

Given an $n$-configuration tableau $\mathfrak{C}=(\lambda_{i,j})$, we write $\gamma_{i,j}$ for a proper walk from cell $\lambda_{i,i}$ to cell $\lambda_{1,j}$; $\Gamma_{i,j}$ denotes the set of all such proper walks $\gamma_{i,j}$. Due to the cells’ indexing of configuration tableaux, since $\gamma_{i,j}$ is proper,

• •

$\gamma_{i,j}$ is a descending walk iff $i\leq j$,

• •

$\gamma_{i,j}$ is a falling walk iff $i\geq j$.

The weight $w(\gamma_{i,j})$ of the walk $\gamma_{i,j}$ is the sum of its cells’ values, i.e.,

[TABLE]

We write $\overline{\omega}_{i,j}(\mathfrak{C})$ for the maximal weight over all walks $\gamma_{i,j}\in\Gamma_{i,j}$, that is

[TABLE]

which determines the function

[TABLE]

We write $\underline{\omega}_{i,j}(\mathfrak{C})$ for the minimal weight over all walks $\gamma_{i,j}\in\Gamma_{i,j}$, that is

[TABLE]

When $i=j$, the set $\Gamma_{i,i}$ consists of exactly one proper walk $\gamma_{i,i}$ which is both a descending walk and a falling walk, and thus (cf. (6.5))

[TABLE]

for any $i=1,\dots,n$. On the other hand, for fixed $i=1$, the weight

[TABLE]

is determined by a unique (bottom) walk, which precisely encodes the first row $\mathfrak{r}_{1}$ of $\mathfrak{C}$. Hence, the weights of the walks $\gamma_{1,j}$ record the full data on $\mathfrak{r}_{1}$.

Let $\Theta_{i,j}(\mathfrak{C})$ be the set of all horizontal covers $\theta_{i,j}$ of columns $1,\dots,j$ by rows $1,\dots,i$ with $i\geq j$, cf.  §6.1. Namely, $\theta_{i,j}$ is a collection of cells $\lambda_{i_{1},1},\dots,\lambda_{i_{j},j}$ in $\mathfrak{C}:=(\lambda_{i,j})$ such that $i_{1}\geq i_{2}\geq\cdots\geq i_{j}$ with $i_{t}\in\{1,\dots,j\}$; in particular $\theta_{i,i}=\lambda_{i,i}$ for each $i=1,\dots,n.$

The weight $w(\theta_{i,j})$ of a cover $\theta_{i,j}$ is the sum of its cells’ values, i.e.,

[TABLE]

Using these weights we introduce a new function on $n$-configuration tableaux:

[TABLE]

defined for $i\geq j$ as

[TABLE]

and $\eta_{i,j}(\mathfrak{C}):=0$ whenever $i<j$. Explicitly, in terms of subsequences, the value of the function $\eta_{i,j}$ can be computed directly as

[TABLE]

In particular, for $i=j$ we have $\eta_{i,i}(\mathfrak{C})=\tau_{i}$, while by the configuration law (6.4) for every $i=1,\dots,n$ we obtain that

[TABLE]

So, we see that $\eta_{i,1}(\mathfrak{C})$, $i=1,\dots,n$, records explicitly the full data on the first column $\mathfrak{c}_{1}$ of $\mathfrak{C}$.

The fixed structure of configuration tableaux enables a straightforward execution of operations that resize the tableaux – shrink or extend them.

Remark 6.12**.**

The deletion of the bottom row or the $n$’th diagonal of an $n$-configuration tableau results in a new proper $(n-1)$-configuration tableau. This does not hold for the deletion of the left column, as seen in Example 6.14 below.

On the other hand, one can enlarge the size of a tableaux through tableau injections, fully preserving its content. The possible injections are determined by column mappings, depending on the size of the target tableau, while row mappings are always one-to-one.

Remark 6.13**.**

Among the possible injections of configuration tableaux of different sizes, we are interested in the right injection, that is the map

[TABLE]

that embeds an $m$-configuration tableau $\mathfrak{C}\in\mathsf{CTab}_{m}$ in the $n$-configuration tableau whose $m\times m$ right part is identically $\mathfrak{C}$ and all its cells in the left columns $1,\dots,n-m$ are of value [math].

6.3. Semi-standard tableaux vs. configuration tableaux

Every semi-standard tableau $\mathfrak{T}\in\mathsf{Tab}_{n}$ is associated to an $n$-configuration tableau $\mathfrak{C}\in\mathsf{CTab}_{n}$ by the map

[TABLE]

defined as

[TABLE]

i.e., the cell $\lambda_{i,\ell}$ of $\mathfrak{C}=(\lambda_{i,j})$ is assigned with the occurrence number of the letter $a_{\ell}$ in the row $\mathfrak{r}_{i}$ of  $\mathfrak{T}$, and $\mathfrak{T}_{0}\longmapsto\mathfrak{C}_{0}$. By this setting the letter $a_{\ell}$ in $\mathfrak{T}$ corresponds to the $\ell$’th diagonal of $\mathfrak{C}$, enumerated starting from bottom left. Accordingly, the $\ell$-trace $\tau_{\ell}$ of $\mathfrak{C}$, cf. (6.5), gives the total occurrences of the letter $\alpha_{\ell}$ in $\mathfrak{T}$, which is read off from the $\ell$’th diagonal of $\mathfrak{C}$.

Example 6.14**.**

Let $w=a_{4}\,a_{3}a_{4}\,{a_{2}}{a_{2}}{a_{4}}{a_{4}}\,{a_{1}}{a_{1}}{a_{1}}{a_{3}}{a_{3}}{a_{4}}$ be a standard tableau for which the map $\mathscr{T}_{\operatorname{ctab}}:\mathfrak{T}_{w}\longmapsto\mathfrak{C}$ is as follows:

[TABLE]

The traces of $\mathfrak{C}$ are then $\tau_{1}(\mathfrak{C})=3$, $\tau_{2}(\mathfrak{C})=2$, $\tau_{3}(\mathfrak{C})=3$, and $\tau_{4}(\mathfrak{C})=5$.

This example also shows that tableaux are not necessarily semi-standard, e.g., see the second column that is not (strictly) increasing.

Remark 6.15**.**

For every semi-standard tableau $\mathfrak{T}\in\mathsf{Tab}_{n}$ assigning (6.13) admits the configuration laws   (6.4) as for every $i<j$ the $j$’th row $\mathfrak{r}_{j}$ of $\mathfrak{T}$ dominates its $i$’th row $\mathfrak{r}_{i}$.

An important property of the map (6.12) is that it also precisely records the shape of semi-standard tableaux. That is, the shape $(\lambda_{1},\dots,\lambda_{m})$ of a tableau $\mathfrak{T}$ is encoded in its image $\mathscr{T}_{\operatorname{ctab}}(\mathfrak{T})$ as the sums of cells’ values of the rows:

[TABLE]

In general, an $n$-configuration tableau records all the information that its pre-image semi-standard tableau carries, in a numerical way.

Theorem 6.16**.**

The map $\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ in (6.12) is bijective.

Proof.

Let $\mathfrak{r}_{i}=\alpha_{\ell_{1}}^{q_{1}}\alpha_{\ell_{2}}^{q_{2}}\cdots\alpha_{\ell_{m}}^{q_{m}}$, where $\ell_{1},\dots,\ell_{m}\in L\subseteq\{i,\dots,n\}$ and $q_{t}\in\mathbb{N}$, be the $i$’th row of the tableau $\mathfrak{T}\in\mathsf{Tab}(\mathcal{A}_{n})$. Then  $\mathfrak{r}_{i}$ can be rewritten uniquely as $\alpha_{i}^{q_{i}}\alpha_{i+1}^{q_{i+1}}\cdots\alpha_{n}^{q_{n}}$ with $q_{t}\in\mathbb{N}_{0}$ for every $\ell_{t}\in\{i,\dots,m\}$ and $q_{t}=0$ when $t\in\{i,\dots,n\}\setminus L.$ Thus, the row mapping

[TABLE]

is injective, i.e., $\lambda_{i,t}=q_{t}$, $t=i,\dots,n$. As this holds for the restriction $\mathscr{T}_{\operatorname{ctab}}|_{\mathfrak{r}_{i}}$ of $\mathscr{T}_{\operatorname{ctab}}$ to any row $\mathfrak{r}_{i}$, and $\mathscr{T}_{\operatorname{ctab}}$ maps row-to-row, we deduce that $\mathscr{T}_{\operatorname{ctab}}$ is injective over the whole tableau (and also respects the configuration laws (6.4) by Remark 6.15).

To see that $\mathscr{T}_{\operatorname{ctab}}$ is surjective, use ($*$ ‣ 6.3) to reproduce each row $\mathfrak{r}_{i}$ of $\mathfrak{T}$ from the $i$’th row of $\mathfrak{C}$, which together, due to the configuration laws (6.4), obey the dominance relations in semi-standard tableaux. ∎

The next conclusions are now immediate.

Corollary 6.17**.**

The restriction

[TABLE]

of $\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ to standard tableaux is bijective.

In view of Theorem 6.16, and (6.1), we define the map

[TABLE]

and write

[TABLE]

for short. For the empty word $e\in\mathcal{A}_{n}^{*}$, we formally set $\mathfrak{C}_{e}:=\mathfrak{C}_{0}$ to be the null configuration tableau $\mathfrak{C}_{0}$ (Definition 6.11).

Corollary 6.18**.**

The map

[TABLE]

is bijective, and $u\equiv_{\operatorname{plc}}v$ iff $\mathfrak{C}_{u}=\mathfrak{C}_{v}$, for any $u,v,w\in\mathcal{A}_{n}^{*}$.

Proof.

Compose Theorem 6.16 with Theorem 6.5. ∎

Similar to Remark 6.6, by the latter corollary, the map $\mathscr{P}_{\operatorname{ctab}}:\mathcal{P}_{n}\longrightarrow\mathsf{CTab}_{n}$ can be realized as a monoid homomorphism, where $\mathsf{CTab}_{n}$ is a monoid whose operation (insertion by concatenating) is induced by the next algorithm. This view is compatible with the monoid structure of $\mathsf{Tab}_{n}$ via the bijection $\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ – reads now a tableau isomorphism.

Algorithm 6.19** (Encoding Algorithm).**

To encode a word $w\in\mathcal{A}_{n}^{+}$ in an $n$-configuration tableau $\mathfrak{C}$, start with an empty $n$-configuration tableau $\mathfrak{C}:=\mathfrak{C}_{e}$ and perform the following letter by letter.

To encode the letter $a_{\ell}\in\mathcal{A}_{n}$ in $\mathfrak{C}$, start with the first row $i=1$

• •

incitement $\lambda_{i,\ell}$ by $1$, i.e., $\lambda_{i,\ell}\leftarrowtail\lambda_{i,\ell}+1$;

• •

decrement $\lambda_{i,k_{i}}$ by $1$, where $k_{i}>\ell$ is the minimal index with $\lambda_{i,k_{i}}>0$.

Repeat this same procedure to insert $a_{k_{i}}$ to row $i+1$ of $\mathfrak{C}$, as long as $i<n$.

This encoding of a word $w$ in $\mathfrak{C}$ is denoted by $w\looparrowright\mathfrak{C}$, resulting in the configuration tableau $\mathfrak{C}_{w}$.

By this algorithm we see that the restriction of $\mathcal{A}^{+}_{[1:n]}$ to $\mathcal{A}^{+}_{[\ell:n]}$, i.e., to words over the convex sub-alphabet $\mathcal{A}_{[\ell:n]}:=\{a_{\ell},\dots,a_{n}\}\subseteq_{\operatorname{cx}}\{a_{1},\dots,a_{n}\}$, reads in terms of configuration tableaux as the image of right injection $\operatorname{Inj_{r}}:\mathsf{CTab}_{n-\ell+1}\lhook\joinrel\longrightarrow\mathsf{CTab}_{n}$, cf. Remark 6.13.

Due to the special structure of configuration tableaux, provided by the configuration laws  (6.4), we obtain the next important property.

Observation 6.20**.**

The bijection $\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ (Theorem 6.16), together with our construction, provides the equality

[TABLE]

for any $w\in\mathcal{A}_{n}^{*}$, cf. (6.3) and (6.6). Namely, weights of descending walks $\gamma_{i,j}$ in a configuration tableaux  $\mathfrak{C}_{w}$ are translated to subwords over the convex sub-alphabet $\mathcal{A}_{[i:j]}\subset\mathcal{A}_{[1:n]}=\mathcal{A}_{n}$ in $\mathfrak{T}_{w}$, and vice versa. That is, the weight $\overline{\omega}_{i,j}(\mathfrak{C}_{w})$ gives the length of longest nondecreasing subword in $\mathfrak{T}_{w}$ with letters in $\mathcal{A}_{[i:j]}$.

The weight of a descending walk $\gamma_{i,j}$ from cell $(i,i)$ to cell $(1,j)$ in $\mathfrak{C}\in\mathsf{CTab}_{n}$, $j\geq i$, is greater than the weight of every descending walk $\gamma_{k,j}$ from $(k,k)$ to $(1,j)$ for $k\geq i$, i.e., $w(\gamma_{i,j})\geq w(\gamma_{k,j})$ for any $i\leq k\leq j$. Indeed, any such descending walk $\gamma_{k,j}$ starts with a sub-row whose cells lay above those of $\gamma_{i,j}$, which by the configuration law (6.4) has a lower weight, and this argument applies inductively.

[TABLE]

(This property fits well with Theorem 6.4, as configuration tableaux record numerically Young tableaux.)

6.4. Reversal of words

Recall that the reversal $\overleftarrow{w}$ of a word $w=\overrightarrow{w}$ is the rewriting of $w$ from right to left (Definition 1.5). Tableaux of reversals of words in general do not preserve the equivalence relation $\equiv_{\operatorname{tab}}$, in sense that we may have ${\operatorname{tab}}(u)\neq{\operatorname{tab}}(v)$ with ${\operatorname{tab}}(\overleftarrow{u})={\operatorname{tab}}(\overleftarrow{v})$, or vice versa. The same may happen for configuration tableaux, cf. Theorem 6.16.

Example 6.21**.**

For example, let $w=bc\,aabc$, $u=bcc\,aab$ and $v=c\,bc\,aab$, for which $\mathfrak{T}_{u}\neq\mathfrak{T}_{v}$, i.e., $u\not\equiv_{\operatorname{tab}}v$. But the tableau $\mathfrak{T}_{w}$ of $w$ is the same as the tableau of the reversal of both $u$ and $v$, that is $\mathfrak{T}_{w}={\operatorname{tab}}(\overleftarrow{u})={\operatorname{tab}}(\overleftarrow{v}).$

Howeverm for the case of standard tableaux we do have the following useful correspondence.

Proposition 6.22**.**

Suppose $\mathfrak{T}_{u}={\operatorname{tab}}(v)$ and $\mathfrak{T}_{v}={\operatorname{tab}}(v)$ are standard tableaux in $\mathsf{STab}_{n}$, then

[TABLE]

where $\mathfrak{T}^{\prime}_{u}={\operatorname{tab}}(\overleftarrow{u})$ and $\mathfrak{T}^{\prime}_{v}={\operatorname{tab}}(\overleftarrow{v})$.

Proof.

Let $\mathfrak{T}^{\operatorname{t}}_{u}$ and $\mathfrak{T}^{\operatorname{t}}_{v}$ be the transpose tableaux of $\mathfrak{T}_{u}$ and $\mathfrak{T}_{v}$, respectively. Then $\mathfrak{T}_{u}^{\operatorname{t}}={\operatorname{tab}}(\overleftarrow{u})$ and $\mathfrak{T}_{v}^{\operatorname{t}}={\operatorname{tab}}(\overleftarrow{v})$, by [44, Theorem 3.2.3], and the assertion follows at once. ∎

In comparison to the case of standard tableaux, due to their configuration laws, standard configuration tableaux do not always admit transposition in classical terms.

Example 6.23**.**

The word $\overrightarrow{u}=bdac$ and its reversal $\overleftarrow{u}=cadb$ are expressed by configuration tableaux as

[TABLE]

The reversal of $\overrightarrow{v}=dbac$ is $\overleftarrow{v}=cabd$, and they described in terms of configuration tableaux as

[TABLE]

Thus, $u\neq v$ with ${\operatorname{ctab}}(u)={\operatorname{ctab}}(v)$, but ${\operatorname{ctab}}(\overleftarrow{u})\neq{\operatorname{ctab}}(\overleftarrow{v})$.

So we see that reversing of words is not translated to transposition of standard configuration tableaux, yet we can employ configuration tableaux for the study of reversed words.

Lemma 6.24**.**

Suppose $\mathfrak{C}_{w}={\operatorname{ctab}}(w)$ is a standard $n$-configuration tableau in $\mathsf{SCTab}_{n}$ with nonzero cells $\lambda_{i,k}=1$ and $\lambda_{j,k+1}=1$, where $i<j$. Then, for the nonzero cells $\lambda^{\prime}_{i^{\prime},k}$ and $\lambda^{\prime}_{j^{\prime},k+1}$ of $\mathfrak{C}_{w}^{\prime}={\operatorname{ctab}}(\overleftarrow{w})$, we have $j^{\prime}\leq i^{\prime}$.

Proof.

Since $j>i$ where $\lambda_{i,k}=1$ and $\lambda_{j,k+1}=1$, the translation of $\mathfrak{C}_{w}={\operatorname{ctab}}(w)$ to $\mathfrak{T}_{w}={\operatorname{tab}}(w)$ by Theorem 6.16 means that in $\mathfrak{T}_{w}$ the letter $a_{k+1}$ is on the left to $a_{k}$, which implies that $a_{k+1}$ is on the right to $a_{k}$ in $\mathfrak{T}_{w}^{\prime}={\operatorname{tab}}(\overleftarrow{w})$ by Proposition 6.22. Then, the one-to-one correspondence $\mathscr{T}_{\operatorname{ctab}}:\mathfrak{T}_{w}^{\prime}\longmapsto\mathfrak{C}_{w}^{\prime}$,by Theorem 6.16, gives the required. ∎

Now we are first able to directly tied the underlying congruence $\equiv_{\operatorname{clk}}$ of the cloaktic monoid (Definition  5.1) to tableau equivalence $\equiv_{\operatorname{tab}}$ in the class of standard tableaux.

Theorem 6.25**.**

Let $\mathfrak{T}_{u}={\operatorname{tab}}(u)$ and $\mathfrak{T}_{v}={\operatorname{tab}}(v)$ be standard tableaux in $\mathsf{STab}_{n}$, with $u,v\in\mathcal{A}^{+}_{n}$, then

[TABLE]

where $\mathfrak{T}^{\prime}_{u}={\operatorname{tab}}(\overleftarrow{u})$ and $\mathfrak{T}^{\prime}_{v}={\operatorname{tab}}(\overleftarrow{v})$.

Proof.

Recall that by definition $u\equiv_{\operatorname{tab}}v$ iff $\mathfrak{T}_{u}=\mathfrak{T}_{v}$. The part $(\Leftrightarrow\mathfrak{T}^{\prime}_{u}=\mathfrak{T}^{\prime}_{v})$ has already been proven in Proposition 6.22, which also asserts that $u\equiv_{\operatorname{clk}}v$ iff $\mathfrak{T}_{u}\equiv_{\operatorname{clk}}\mathfrak{T}_{v}$. We prove the rest. $(\Rightarrow)$: Immediate since $\equiv_{\operatorname{tab}}$ implies $\equiv_{\operatorname{clk}}$, by Proposition 6.9. $(\Leftarrow)$: Let $\mathfrak{C}_{u}={\operatorname{ctab}}(u)$, $\mathfrak{C}_{v}={\operatorname{ctab}}(v)$, $\mathfrak{C}^{\prime}_{u}={\operatorname{ctab}}(\overleftarrow{u})$, and $\mathfrak{C}^{\prime}_{v}={\operatorname{ctab}}(\overleftarrow{v})$, and denote their cells respectively by $[\lambda_{u}]_{i,j}$, $[\lambda_{v}]_{i,j}$, $[\lambda^{\prime}_{u}]_{i,j}$, and $[\lambda^{\prime}_{v}]_{i,j}$. Assume that $\mathfrak{T}_{u}\neq\mathfrak{T}_{u}$, and thus $\mathfrak{T}^{\prime}_{u}\neq\mathfrak{T}^{\prime}_{v}$ by Proposition  6.22, implying that $\mathfrak{C}_{u}\neq\mathfrak{C}_{v}$ and $\mathfrak{C}^{\prime}_{u}\neq\mathfrak{C}^{\prime}_{v}$, by Corollary  6.17. Recall that each diagonal of a standard $n$-configuration tableau has a unique nonzero cell of value $1$.

Proof by induction on $n$. The cases of $n=1$ and $n=2$ are clear. Assuming the implication holds for $\mathcal{A}_{n-1}\subseteq_{\operatorname{cx}}\mathcal{A}_{n}$, the only change happens by including the extra letter $a_{n}$ is at the $n^{\prime}$th diagonal of standard $n$-configuration tableaux. Then, by the induction assumption, $\mathfrak{C}_{u}$ and $\mathfrak{C}_{v}$ have the same $(n-1)$’th diagonal whose nonzero cell is say at $\lambda_{k,n-1}=1$.

Suppose that $[\lambda_{u}]_{i,n}=1$ and $[\lambda_{v}]_{j,n}=1$, say for $j>i$, and assume that $u\equiv_{\operatorname{clk}}v$ (or equivalently $\mathfrak{T}_{u}\equiv_{\operatorname{clk}}\mathfrak{T}_{v}$), which in terms of the function (6.3) implies that the equality $\natural_{[s:t]}(\mathfrak{T}_{u})=\natural_{[s:t]}(\mathfrak{T}_{v})$ holds for all $1\leq s\leq t\leq n$. We deduce that $j>i>k$, since otherwise we would get $\natural_{[i:n]}(\mathfrak{T}_{u})>\natural_{[i:n]}(\mathfrak{T}_{v})$.

Similarly, consider the $n$-configuration tableaux $\mathfrak{C}^{\prime}_{u}$ and $\mathfrak{C}^{\prime}_{v}$ of $\overleftarrow{u}$ and $\overleftarrow{v}$ respectively, and let $[\lambda^{\prime}_{u}]_{k^{\prime},n-1}$, $[\lambda^{\prime}_{u}]_{i^{\prime},n},$ and $[\lambda^{\prime}_{v}]_{j^{\prime},n}$ be respectively the nonzero cells of $\mathfrak{C}^{\prime}_{u}$ and $\mathfrak{C}^{\prime}_{v}$, say with $i^{\prime}>j^{\prime}$. Then, assuming that $\overleftarrow{u}\equiv_{\operatorname{knu}}\overleftarrow{v}$, by the same argument as above we obtain $i^{\prime}>j^{\prime}>k^{\prime}$, since otherwise we would have $\natural_{[j^{\prime}:n]}(\mathfrak{C}^{\prime}_{v})>\natural_{[j^{\prime}:n]}(\mathfrak{C}^{\prime}_{u})$. But, since $j>i>k$, by Lemma 6.24 we should have $j^{\prime}\leq k^{\prime}$ and $i^{\prime}\leq k^{\prime}$ – a contradiction. ∎

7. Representations of tableaux and of the plactic monoid

In this section we utilize the tropical representations of the cloaktic monoid (§5.1) and the co-cloaktic monoid (§5.3), which essentially record lengths of longest subwords over convex sub-alphbets, to construct linear representations of semi-standard tableaux. Linear representations of the plactic monoid $\mathsf{PLC}_{n}$ are then follow from the correspondence between the elements of $\mathsf{PLC}_{n}$ and $n$-configuration tableaux (Corollary  6.18). The latter correspond uniquely to semi-standard tableaux via the map \mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}\mathsf{CTab}_{n} (Theorem 6.16). Our next step is to establish the two maps

[TABLE]

from $n$-configuration tableaux to tropical matrices. Then their compositions with the bijection $\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ provide the maps

[TABLE]

At this point, the digraph realization of tropical matrices (cf. §3.2) is of major importance. To make our image matrices more comprehensible, in what follows, for simplicity, we set the formal variable $\kappa$ in the troplacitic matrix algebra $\mathfrak{A}_{n}$ to have fixed value $\kappa:=1$, cf. (4.4).

The study in this section is accompanied with a collection of pathological examples that demonstrate the difficulties towards faithfully representing tableaux. Our development is supported by the use of configuration tableaux that enables an easier analysis and helps to better understanding the combinatorial arguments. But before that, in order to complete these representations, we need another crucial competent.

7.1. The co-plactic monoid

Co-mirroring of words (Definition 1.6) leads to the following monoid structure, drown from the plactic monoid $\mathsf{PLC}_{n}$ (Definition 2.1). Recall that $\equiv_{\operatorname{plc}}$ is an additional notation for the underlying congruence  $\equiv_{\operatorname{knu}}$ of $\mathsf{PLC}_{n}$.

Definition 7.1**.**

The co-plactic monoid is the monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{P}}}_{n}:={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{PLC}}}(\mathcal{A}_{n})$ generated by a finite ordered set of elements $\mathcal{A}_{n}:=\{a_{1},\dots,a_{n}\}$, subject to the equivalence relation ${\equiv^{\operatorname{co}}_{\operatorname{plc}}}$, defined as

[TABLE]

Namely ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{P}}}_{n}:=\mathcal{A}_{n}^{*}/_{{\equiv^{\operatorname{co}}_{\operatorname{plc}}}}$. We say that ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{PLC}}}_{n}$ is of rank $n$, and write ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{PLC}}}_{n}$ for ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathsf{PLC}}}(\mathcal{A}_{n})$ when $\mathcal{A}_{n}$ is arbitrary.

In other words, the relation ${\equiv^{\operatorname{co}}_{\operatorname{plc}}}$ is satisfied if the words $a^{\prime}_{\ell}:={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}_{n}(a_{\ell})$ in $\mathcal{A}_{n}^{*}$, (lexicographically) ordered as $a_{1}^{\prime}<a^{\prime}_{2}<\cdots<a^{\prime}_{n}$ (cf. Remark 1.7), admit the Knuth relations (KNT). Note that here we initially consider only finitely generated monoid, to have the co-mirroring map well defined.

Theorem 7.2**.**

The co-mirrors (1.4) of the generators $a_{1},\dots,a_{n}$ of the plactic monoid $\mathcal{P}_{n}$,

[TABLE]

ordered as $a^{\prime}_{1}<a^{\prime}_{2}<\cdots<a^{\prime}_{n},$ admit the Knuth relations (KNT) and thus the congruence $\equiv_{\operatorname{plc}}$ of $\mathcal{P}_{n}$ implies the equivalence ${\equiv^{\operatorname{co}}_{\operatorname{plc}}}$.

Configuration tableaux correspond bijectively to elements of $\mathcal{P}_{n}$ (Corollary 6.18) and are utilized to prove the theorem by a straightforward computation, heavily based on the Encoding Algorithm 6.19. The proof is performed by induction on $n$, where the induction step assumes the implication for $\{a_{2},\dots,a_{n}\}$, which corresponds to assuming the implication for the upper $n-1$ rows of an $n$-configuration tableau. The proof is fairly technical, including several cases, and appears in its full details in Appendix A, together with some additional examples.

Recall from Remark 1.7 that ${\mathcal{A}^{*}_{n}}^{[1]}$ is the finitely monoid generated by ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}(a_{1}),\dots,{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}(a_{n})$, and associated with the homomorphism $\partial_{1}:\mathcal{A}^{*}_{n}\longrightarrow{\\ tA^{*}_{n}}^{[1]}$.

Corollary 7.3**.**

The congruence $\equiv_{\operatorname{plc}}$ implies the equivalence ${\equiv^{\operatorname{co}}_{\operatorname{plc}}}$ (which by itself is then a congruence), the map

[TABLE]

is a monoid isomorphism, and the plactic monoid $\mathcal{P}_{n}:=\mathcal{A}^{*}_{n}/_{\equiv_{\operatorname{plc}}}$ of rank $n$ is also a co-plactic monoid ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{P}}}_{n}$ of rank $n$ for which the map

[TABLE]

is a surjective homomorphism.

Corollary 7.4**.**

The plactic monoid $\mathcal{P}_{n}=\langle\,{a_{1},\dots,a_{n}}\,\rangle$ contains the inductive chain of plactic submonoids

[TABLE]

of rank $n$, generated by $a_{1}^{[i]}={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}_{n}(a_{1}^{[i-1]}),\dots,a_{n}^{[i]}={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}_{n}(a_{n}^{[i-1]})$. Furthermore, the maps

[TABLE]

cf. Remark 1.7, are monoid isomorphisms.

Remark 7.5**.**

The surjective homomorphism $\eth:\mathcal{P}_{n}\;{\relbar\joinrel\twoheadrightarrow}\;\mathcal{K}_{n}$ (Corollary 6.10) induces the surjective homomorphism ${{}^{\operatorname{co}}\joinrel\,\joinrel{\,\eth}}:\mathcal{P}_{n}\longrightarrow{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ via the diagram

[TABLE]

where $\mathcal{K}_{n}$ and ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ are linearly represented by $\mathfrak{A}^{\times}_{n}$ and ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$, respectively.

7.2. Representations of configuration tableaux

We start by defining our first map

[TABLE]

that sends an $n$-configuration tableau $\mathfrak{C}=(\lambda_{i,j})$ to the matrix $U=(u_{i,j})$ determined by

[TABLE]

where $\overline{\omega}_{i,j}$ is given by (6.6) in §6.3. This map is then realized as a monoid homomorphism.

Lemma 7.6**.**

The matrix $W=\mathscr{C}_{\operatorname{mat}}(\mathfrak{C})$ indeed belongs to $\mathfrak{A}^{\times}_{n}$, cf. (4.4), for any $\mathfrak{C}\in\mathsf{CTab}_{n}$.

Proof.

$\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ is bijective (Theorem 6.16), with \natural_{[i:j]}\big{(}\mathfrak{T}_{w}\big{)}=\overline{\omega}_{i,j}(\mathfrak{C}_{w}) by Observation  6.20, while $\mathsf{Tab}_{n}$ and $\mathsf{CTab}_{n}$ are isomorphic (as monoids) to the plactic monoid $\mathcal{P}_{n}$ (Remark  6.6 and Corollary 6.18, respectively). Furthermore, the map $\eth:\mathcal{P}_{n}\longrightarrow\mathcal{K}_{n}$, $[w]_{\operatorname{plc}}\mapsto[w]_{\operatorname{clk}},$ is a surjective monoid homomorphism (Corollary 6.10), while $\mho:\mathcal{K}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$, $a_{\ell}\mapsto{A}_{(\ell)}$ for $\ell=1,\dots,n$, is an isomorphism (Theorem 5.6). Thus we only need to show that $\mathscr{C}_{\operatorname{mat}}$ maps the $n$-configuration tableaux assigned to letters $a_{\ell}$ in   $\mathcal{A}_{n}$ to the generating matrices ${A}_{(\ell)}$ of  $\mathfrak{A}^{\times}_{n}$, namely that $\mathscr{C}_{\operatorname{mat}}(\mathfrak{C}_{a_{\ell}})={A}_{(\ell)}$ for each $\ell=1,\dots,n$.

Drawing the $n$-configuration tableau of a letter $a_{\ell}$, which is of the form

[TABLE]

by the definition of descending walks (§6.1) and their weights (6.6) we observe that

1. (i)

$\overline{\omega}_{i,j}(\mathfrak{C}_{a_{\ell}})=0$ for $i\leq j<\ell$; 2. (ii)

$\overline{\omega}_{i,j}(\mathfrak{C}_{a_{\ell}})=1$ for $i\leq\ell\leq j$; 3. (iii)

$\overline{\omega}_{i,j}(\mathfrak{C}_{a_{\ell}})=0$ for $\ell<i\leq j$.

Accordingly, the matrix $\mathscr{C}_{\operatorname{mat}}(\mathfrak{C}_{a_{\ell}})$ defined by (7.4) is precisely the triangular matrix ${A}_{(\ell)}:={F}_{(\ell)}\vee E$ defined in  (4.1) and (4.4), with $\kappa=1$ and $E$ as in (4.3). ∎

By Remark 6.6 we conclude that:

Theorem 7.7**.**

$\mathscr{C}_{\operatorname{mat}}:\mathsf{CTab}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$, given in (7.4), is a surjective map, realized a as monoid homomorphism.

Proof.

We know that $\mathscr{P}_{\operatorname{ctab}}:\mathcal{P}_{n}\longrightarrow\mathsf{CTab}_{n}$ is a bijection (Corollary 6.18) realized as a monoid homomorphism, $\eth:\mathcal{P}_{n}\longrightarrow\mathcal{K}_{n}$ is a surjective monoid homomorphism (Corollary 6.10), and $\mho:\mathcal{K}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$ is an isomorphism (Theorem 5.6), thus the diagram

[TABLE]

commutes by Lemma 7.6. ∎

Example 7.8**.**

Applying the map $\mathscr{C}_{\operatorname{mat}}$ in (7.3) to the $3$-configuration tableau $\mathfrak{C}_{w}$ with $w$ the word in Example 6.14, we have

[TABLE]

where $\mathfrak{T}_{w}$ is the Young tableau of $w$.

In terms of configuration tableaux, and the map $\mathscr{C}_{\operatorname{mat}}:\mathsf{CTab}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$, Example 5.10 now reads as follows.

Example 7.9**.**

The representation $\mathscr{C}_{\operatorname{mat}}:\mathsf{CTab}_{3}\longrightarrow\mathfrak{A}^{\times}_{3}$ of $3$-configuration tableaux assigned to the three letter alphabet $\mathcal{A}_{3}=\langle\,{a,b,c}\,\rangle$ is determined by the generators’ mapping

[TABLE]

The Knuth-equivalence (KNT) for triplets in $\mathcal{A}_{3}$ are described in $\mathsf{CTab}_{3}$ and $\mathfrak{A}^{\times}_{3}$ as

[TABLE]

while

[TABLE]

For the increasing word $abc\in\mathcal{A}_{3}^{+}$ we get

[TABLE]

and for the decreasing word $cba\in\mathcal{A}_{3}^{+}$ we have

[TABLE]

For a generic 3-letter tableau $\mathfrak{T}_{w}:=c^{k_{3}}\,b^{j_{2}}c^{k_{2}}\,a^{i_{1}}b^{j_{1}}c^{k_{1}}$, by a direct computation, we obtain

[TABLE]

Then taking $k_{2}^{\prime},k_{2}^{\prime\prime}$ such that $k_{2}^{\prime},k_{2}^{\prime\prime}\leq j_{2}$ and $k^{\prime}_{2}k_{3}^{\prime}=k^{\prime\prime}_{2}k_{3}^{\prime\prime}$, we can produce two different tableaux with a same image, what shows that in general the map $\mathscr{C}_{\operatorname{mat}}$ not injective.

Returning to our running example (cf. Example 6.8), it gives an explicit numerically example for the non-injectivity of the map $\mathscr{C}_{\operatorname{mat}}:\mathsf{Tab}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$.

Example 7.10**.**

Taking the explicit words of Example 6.8, we have

[TABLE]

where on the other also

[TABLE]

Hence $\mathscr{C}_{\operatorname{mat}}(\mathfrak{C}_{u})=\mathscr{C}_{\operatorname{mat}}(\mathfrak{C}_{v})$, but $u\not\equiv_{\operatorname{tab}}v$, or equivalently $u\not\equiv_{\operatorname{plc}}v$ by Theorem 6.5. Namely, the map $\mathscr{C}_{\operatorname{mat}}:\mathsf{CTab}_{3}\longrightarrow\mathfrak{A}^{\times}_{3}$ is not injective

Nevertheless, we do have a partial (row) injection:

Lemma 7.11**.**

The restriction

[TABLE]

of the map $\mathscr{C}_{\operatorname{mat}}$ in (7.3) to the bottom row is a bijective map.

Proof.

By Theorem 6.16, $\operatorname{row}_{1}(\mathfrak{C})$ corresponds uniquely to a nondecreasing word in $\mathcal{A}_{n}^{+}$, which is mapped bijectively to $\operatorname{row}_{1}(W)$ of $W\in\mathfrak{A}^{\times}_{n}$ by Lemma 5.9. ∎

7.3. Co-representations of configuration tableaux

We now turn to our second map

[TABLE]

which we called a co-representation, that uses as its target the matrix co-monoid ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$, cf. (5.9). This map sends an $n$-configuration tableau $\mathfrak{C}\in\mathsf{CTab}_{n}$ to the matrix $V=(v_{i,j})$ defined by

[TABLE]

where $i^{\prime}=n-i+1$, $j^{\prime}=n-j+1$, and $\eta_{i^{\prime},j^{\prime}}$ is the function given by (6.11) in §6.3. The map $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}$ is then realized as a monoid homomorphism. (Recall that, by Theorem 5.20, ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ is isomorphic to the co-cloaktic monoid  ${{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$.)

Lemma 7.12**.**

The matrix $V=\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}(\mathfrak{C})$ indeed belongs to ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ for any $\mathfrak{C}\in\mathsf{CTab}_{n}$.

Proof.

We follow similar arguments as in the proof of Lemma 7.6, with an additional step. First, $\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ is a bijection (Theorem 6.16) with $\mathsf{Tab}_{n}$ and $\mathsf{CTab}_{n}$ isomorphic (as monoids) to the plactic monoid $\mathcal{P}_{n}$ (Theorem 6.16, Remark  6.6, and Corollary 6.18, respectively). On the other hand, $\eth:\mathcal{P}_{n}\longrightarrow\mathcal{K}_{n}$, $[w]_{\operatorname{plc}}\mapsto[w]_{\operatorname{clk}},$ is a surjective homomorphism (Corollary 6.10), while $\mho:\mathcal{K}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$, $a_{\ell}\mapsto{A}_{(\ell)}$ is an isomorphism (Theorem 5.6). Finally we have the surjective homomorphism $\Game:\mathfrak{A}^{\times}_{n}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}},$ ${A}_{(\ell)}\mapsto{\widecheck{A}}_{(\ell)}$, cf. (5.12). Thus we only need to show that $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}$ maps each $n$-configuration tableau assigned to the letter $a_{\ell}$ in $\mathcal{A}_{n}$ to the matrix ${A}_{(\ell)}$ of $\mathfrak{A}^{\times}_{n}$, namely that $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}(\mathfrak{C}_{a_{\ell}})={\widecheck{A}}_{(\ell)}$.

Observing the $n$-configuration tableau $\mathfrak{C}_{a_{\ell}}$ assigned to the letter $a_{\ell}$, see (7.5), by the definition of horizontal covers and (6.11) we obtain that:

1. (i)

$\eta_{i,j}(\mathfrak{C})=1$ for $i=j=\ell$; 2. (ii)

$\eta_{i,j}(\mathfrak{C})=0$ otherwise.

Then the image matrix $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}(\mathfrak{C}_{a_{\ell}})$ defined by (7.8) is precisely the triangular matrix ${\widecheck{A}}_{(\ell)}:={F}_{(\ell)}\vee E$ in  (5.6), with $\kappa=1$ and $\mathbb{1}=0$, as usual. ∎

Theorem 7.13**.**

$\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}:\mathsf{CTab}_{n}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$, given by (7.8), is a surjective map, realized a as monoid homomorphism.

Proof.

We know that $\mathscr{P}_{\operatorname{ctab}}:\mathcal{P}_{n}\longrightarrow\mathsf{CTab}_{n}$ is a bijection (Corollary 6.18) realized as a monoid homomorphism, $\partial:\mathcal{P}_{n}\;{\relbar\joinrel\twoheadrightarrow}\;{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{P}}}_{n}$ is a surjective homomorphism (Corollary 7.3), ${{}^{\operatorname{co}}\joinrel\,\joinrel{\,\eth}}:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{P}}}_{n}\longrightarrow{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ is a surjective homomorphism induced from $\eth:{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{P}}}_{n}\longrightarrow{{}^{\operatorname{co}}\joinrel\,\joinrel{\mathcal{K}}}_{n}$ (Corollary 6.10), and \Omega:\mathcal{K}_{n}\overset{\,\smash{\raisebox{-1.93747pt}{\scriptstyle\sim}}\,}{\;{\relbar\mathrel{\mkern-6.0mu}\relbar\mathrel{\mkern-6.0mu}\rightarrow}\;}{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}} is an isomorphism (Theorem 5.20). Hence, by Lemma 7.12, the diagram

[TABLE]

commutes. ∎

In comparison to the representation $\mathscr{C}_{\operatorname{mat}}:\mathsf{CTab}_{3}\longrightarrow\mathfrak{A}^{\times}_{3}$ in Example 7.9, for the co-representation $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}:\mathsf{CTab}_{3}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ we have the following.

Example 7.14**.**

The $3$-configuration tableaux corresponding to the three letter alphabet $\mathcal{A}_{3}=\langle\,{a,b,c}\,\rangle$ are co-represented by $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}$ in ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ by the matrices

[TABLE]

The Knuth-equivalence (KNT) for triplets in $\mathcal{A}_{3}$ are described in ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ as

[TABLE]

while

[TABLE]

For the increasing word $abc\in\mathcal{A}_{3}^{*}$ we get

[TABLE]

and for the decreasing word $cba\in\mathcal{A}_{3}^{*}$ we have

[TABLE]

But

[TABLE]

where we also have

[TABLE]

which shows that in general the co-representation $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}:\mathsf{CTab}_{n}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ is not injective.

For a general 3-letter tableau $\mathfrak{T}_{w}:=c^{k_{3}}\,b^{j_{2}}c^{k_{2}}\,a^{i_{1}}b^{j_{1}}c^{k_{1}}$, by a direct computation, we have

[TABLE]

showing that the values $j_{1},k_{2}$ make the map $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}$ non-injective. Recall that in Example 7.9, the cause for non-injectivity was $j_{2},k_{2}$.

Yet, in other situations, the use of co-representations $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}:\mathsf{CTab}_{n}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ can be useful.

Example 7.15**.**

Applying the co-representation $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}:\mathsf{CTab}_{3}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ to the $3$-configuration tableaux in Example 7.10, we have

[TABLE]

where on the other

[TABLE]

Hence $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}(\mathfrak{C}_{u})\neq\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}(\mathfrak{C}_{v})$, with $u\not\equiv_{\operatorname{tab}}v$. Recall from Example 7.10 that for these tableaux we had $\mathscr{C}_{\operatorname{mat}}(\mathfrak{C}_{u})=\mathscr{C}_{\operatorname{mat}}(\mathfrak{C}_{v})$ which has shown that $\mathscr{C}_{\operatorname{mat}}:\mathsf{CTab}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$ is not injective.

Corollary 7.16**.**

$u\equiv_{\operatorname{knu}}v$* implies both $u\equiv_{\operatorname{clk}}v$ and $u{\equiv^{\operatorname{co}}_{\operatorname{clk}}}v$.*

Proof.

The implication for the equivalence $\equiv_{\operatorname{clk}}$ has been already proven in Proposition 6.9, while that for the equivalence ${\equiv^{\operatorname{co}}_{\operatorname{clk}}}$ follows from Theorem 7.13. ∎

7.4. Linear representations of the plactic monoid

The representations of $n$-configuration tableaux as constructed previously in §7.2 and §7.3 lead directly to a representation of semi-standard tableaux, via the one-to-one correspondence $\mathscr{T}_{\operatorname{ctab}}:\mathsf{Tab}_{n}\longrightarrow\mathsf{CTab}_{n}$ (Theorem   6.16), and thus to a linear representation of the plactic monoid (cf. Remark  6.6).

Theorem 7.17**.**

The map

[TABLE]

is a surjective monoid homomorphism – a linear representation of the plactic monoid $\mathcal{P}_{n}$.

Proof.

Compose the bijection $\mathscr{P}_{\operatorname{ctab}}:\mathcal{P}_{n}\longrightarrow\mathsf{CTab}_{n}$ (Corollary 6.18) separately with the two surjections $\mathscr{C}_{\operatorname{mat}}:\mathsf{CTab}_{n}\longrightarrow\mathfrak{A}^{\times}_{n}$ given by (7.4) (Theorem 7.7) and $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}:\mathsf{CTab}_{n}\longrightarrow{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{n}}}$ given by (7.8) (Theorem  7.13), realized as monoid homomorphisms by Remark 6.6. ∎

Let $u=bdac$ and $v=dbac$, relating respectively the $4$-configuration tableaux

[TABLE]

or equivalently $\mathfrak{T}_{u}\neq\mathfrak{T}_{v}$, and therefore $u\not\equiv_{\operatorname{plc}}v$. On the other hand, their image under the representation $\wp_{4}:\mathcal{P}_{4}\longrightarrow\mathfrak{A}^{\times}_{4}\times{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{4}}}$ is the same

[TABLE]

which shows that in general $\wp_{n}$ is not injective for $n>3.$ Yet, this representation is faithful for the case of the plactic monoid  $\mathcal{P}_{3}$ of rank $3$.

Theorem 7.18**.**

The homomorphism $\wp_{3}:\mathcal{P}_{3}\longrightarrow\mathfrak{A}^{\times}_{3}\times{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ in (7.9) is an isomorphism – a faithful linear representation of the plactic monoid $\mathcal{P}_{3}$.

Proof.

Proof by a straightforward computation. Suppose $\mathfrak{T}_{w}\in\mathsf{Tab}_{n}$ is a tableau of the form

[TABLE]

corresponding uniquely to the $3$-configuration tableau $\mathfrak{C}_{w}={\operatorname{ctab}}(w)$, by Theorem  6.16. As in Theorem 7.17, define $\wp_{3}$ to have the generators’ mapping

[TABLE]

where $A,B,C\in\mathfrak{A}^{\times}_{3}$ and $\widecheck{A},\widecheck{B},\widecheck{C}\in{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ are described in Example 7.9 and 7.14, respectively.

For simplicity, write $i=i_{1},$ $j=j_{1}j_{2}$, $k=k_{1}k_{2}k_{3}$. The direct computation of the matrix products

[TABLE]

in ${\operatorname{TMat}}(\mathbb{T})$ results in the correspondence

[TABLE]

and we need to prove that $\mathfrak{C}_{w}$ can be recover uniquely from the matrices $X=(x_{i,j})$ and $Y=(y_{i,j})$.

First $i_{1}$, $j_{2}$, and $k_{3}$ can be read off immediately from the right column of $Y$, while $j_{1}$ and $k_{1}$ are extracted recursively from the top row of $X$. So, it remains to compute $j_{2}$ and $k_{2}$.

Since $j_{1}$ and $j_{2}$ have been already recovered, from $x_{2,3}$ we can figure out whether $k_{2}\geq j_{1}$ or not. If $k_{2}\geq j_{1}$, then $x_{2,3}=j_{2}k_{2}$, which provides $k_{2}$. Otherwise $k_{2}<j_{1}$, which implies $-k_{2}>-j_{1}$, and thus $y_{1,2}=(-k_{2})(-k_{3})$, which gives $k_{2}$, as $k_{3}$ is already known. (Note that $k_{2}$ can also be computed as $(-k)/((-k_{1})(-k_{3}))$.) ∎

Corollary 7.19**.**

The plactic monoid $\mathcal{P}_{3}$ admits all the semigroup identities satisfied by the monoid ${\operatorname{TMat}}_{3}(\mathbb{T})$ of triangular tropical matrices, in particular the semigroup identities (3.6) with $n=3$

[TABLE]

by letting $x=uv$ and $y=vu$, for any $u,v\in\mathcal{P}_{3}$, cf. (1.10) in Construction 1.9.

Proof.

$\mathcal{P}_{3}$ is isomorphic to the product $\mathfrak{A}^{\times}_{3}\times{{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ (Theorem 7.18), both $\mathfrak{A}^{\times}_{3}$ and ${{}^{\diamond}\joinrel{\mathfrak{A}^{\times}_{3}}}$ are submonoids of ${\operatorname{TMat}}_{3}(\mathbb{T})$, which by Theorem 3.2 satisfies the identity (3.6) for $n=3$. ∎

The maps $\mathscr{C}_{\operatorname{mat}}$ and $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}$ By our construction, one sees that for any $w\in\mathcal{P}_{n}$ the characters (cf. (4.6) and (4.7))

[TABLE]

and thus, embedding $\wp_{n}(w)$ trivially in $2n\times 2n$ matrix $C_{w}$ with $\mathscr{C}_{\operatorname{mat}}(\mathfrak{C}_{w})$ and $\mathscr{C}_{\operatorname{mat}}^{\operatorname{co}}(\mathfrak{C}_{w})$ as its diagonal blocks, by  (7.9) we have

[TABLE]

Recall that $\chi^{+}$ provides the maximal occurrence of a letter in $w$, cf. §4. Then, $\{C_{w}\ {|}\ w\in\mathcal{A}_{n}^{*}\}\subset{\operatorname{TMat}}_{2n}(\mathbb{T})$ is a multiplicative monoid whose members all have multiplicative trace $\operatorname{mtr}=0$ (Definition  3.4).

7.5. Standard Young tableaux and the symmetric group

The special structure of standard tableaux $\mathsf{STab}_{n}$, via the corresponding standard $n$-configuration tableaux (Lemma 6.24), allows their faithful realization in terms of the tropical representation of the cloaktic monoid $\mathcal{K}_{n}$ (Theorem 5.6). (We use the terminology “realization” as $\mathsf{STab}_{n}$ does not form a monoid, and our maps here are set-theoretical maps.)

Theorem 7.20** (Realization of standard tableaux).**

Writing $\mathfrak{T}_{w}={\operatorname{tab}}(\overrightarrow{w})$ and $\mathfrak{T}^{\prime}_{w}={\operatorname{tab}}(\overleftarrow{w})$ for standard tableaux of $\overrightarrow{w}$ and $\overleftarrow{w}$, i.e., $\overrightarrow{w}$ is a permutation of $a_{1},\dots,a_{n}$, then the map

[TABLE]

is injective.

Proof.

Compose Proposition 6.22 with Theorem 6.25. ∎

By the bijective correspondence of the symmetric group $S_{n}$ to standard Young tableaux [44], we define the set-theoretic map

[TABLE]

induced from (7.10), and thus obtain a tropical matrix realization of $S_{n}$.

A supplementary tropical view to $S_{n}$ is briefly as follows. Let $s_{i}\in S_{n}$ be the simple transposition that permutes $i$ with $i+1$ and leaves the other elements of $\{1,\cdots,n\}$ unchanged. The set $\{s_{1},\dots,s_{n-1}\}$ of all these transpositions (called Coxeter transpositions) generates the symmetric group $S_{n}$, and thus every permutation $\sigma\in S_{n}$ can be written in terms of $s_{i}$’s as a word $s_{i_{1}}s_{i_{2}}\cdots s_{i_{m}}$ with $i_{1},\dots,i_{m}\in\{1,\dots,n-1\}$. To get a reduced form of $\sigma$ we take $s_{i_{1}}s_{i_{2}}\cdots s_{i_{m}}$ with minimal $m$.

Consider the transposition $s_{i}$ as a word over $\{1,\dots,n\}$. It consists of exactly $n$ letters, each of which appears once. Applying the map (7.11) to transpositions $s_{i}$, we obtain the induced (set-theoretic) map

[TABLE]

determined now by the generators’ mapping. This realization has special properties, for example all $\mathscr{Q}_{n}(s_{i})$ are quasi-idempotents, i.e., $\mathscr{Q}_{n}(s_{i})=q(\mathscr{Q}_{n}(s_{i}))^{2}$ with a fixed $q\in\mathbb{R}$, establishing an important linkage to Heacke algebras [16].

8. Remarks and open problems

The first question arises from our results concerns a generalization of Theorem 7.18.

Problem 8.1**.**

Is there a faithful linear representation of the plactic monoid of rank $n$ by triangular (or nonsingular) tropical matrices?

By Theorem 3.2, a positive answer to this question solves the problem (cf. [27]):

Problem 8.2**.**

Does the plactic monoid of rank $n$ satisfy a nontrivial semigroup identity?

This paper shows that the plactic monoid $\mathcal{P}_{3}$ of rank $3$ admits all the semigroup identities satisfied by the monoid ${\operatorname{TMat}}_{3}(\mathbb{T})$ of $3\times 3$ triangular tropical matrices (Corollary  7.19). This naturally leads to the converse question:

Problem 8.3**.**

Do $\mathcal{P}_{3}$ and ${\operatorname{TMat}}_{n}(\mathbb{T})$ satisfy exactly the same identities?

A few fundamental facts on the symmetric group $S_{n}$ are well known:

• •

the irreducible representations of $S_{n}$ are in canonical bijection with partitions $\lambda$ of $n$;

• •

the dimension of the irreducible representation corresponding to a given partition $\lambda$ of $n$ is exactly $d_{\lambda}$ – the number of standard Young tableaux of shape $\lambda$.

Actually, these facts were the original motivation for studying Young tableaux, and many combinatorial results on Young tableaux can be expressed by using representations of $S_{n}$, e.g., see [44].

The bijective correspondence of elements of $S_{n}$ to standard Young tableaux, which in their turn are faithfully realizable in $\mathfrak{A}^{\times}_{n}\times\mathfrak{A}^{\times}_{n}$ (Theorem 5.6), leads to a new approach for studying representations of  $S_{n}$. Hook’s formula computes the numbers $d_{\lambda}$ in combinatorial terms of Young tableaux, these numbers are possibly reflected in their tropical matrix realization.

Problem 8.4**.**

Can the $d_{\lambda}$’s be extracted from tropical realizations in an algebraic way?

Appendix A Proofs and completions to §7.1

In this appendix we bring additional details for §7.1, in particular the full detailed proof of Theorem   7.2. We use the terminology in §1.2.

Example A.1**.**

Let $\mathcal{P}_{3}:=\langle\,{a,b,c}\,\rangle$ be the plactic monoid of rank $3$, for which the co-mirrors (1.4) of generators are

[TABLE]

*ordered as $a^{\prime}<b^{\prime}<c^{\prime}$. Then $a^{\prime},b^{\prime},c^{\prime}$ admit the Knuth relations (KNT) and thus the congruence $\equiv_{\operatorname{plc}}$ of $\mathcal{P}_{3}$ implies the equivalence ${\equiv^{\operatorname{co}}_{\operatorname{plc}}}$. Indeed, for $\operatorname{KN1}$ we have 555The middle terms are presented in their canonical tableau form, that is as elements of the plactic monoid, as discussed in §6.1. *

[TABLE]

and

[TABLE]

when $a^{\prime}<b^{\prime}$.

For the second relation $\operatorname{KN2}$ we have 55footnotemark: 5

[TABLE]

where

[TABLE]

for $b^{\prime}<c^{\prime}$.

For the two other 3-letter words we get

[TABLE]

So we see that the Knuth relations (KNT) are “faithfully” preserved.

Let $a^{\prime}_{\ell}:={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}(a_{\ell})$ be the co-mirror of $a_{\ell}\in\mathcal{A}_{n}$, which is a word in $\mathcal{A}_{n}^{*}$, and set $\ell^{\prime}=n-\ell+1$. The $n$-configuration tableau of $a^{\prime}_{\ell}$ is tableau

[TABLE]

whose $\ell^{\prime}$-diagonal is zero; in the case that $\ell=n$ (i.e., when $\ell^{\prime}=1$) the left column is empty.

Theorem 7.2. The co-mirrors (1.4) of the generators $a_{1},\dots,a_{n}$ of the plactic monoid $\mathcal{P}_{n}$, $a^{\prime}_{\ell}:={{}^{\operatorname{co}}\joinrel\,\joinrel{\mathfrak{M}}}(a_{\ell}),$ $\ell=1,\dots,n,$ ordered as $a^{\prime}_{1}<a^{\prime}_{2}<\cdots<a^{\prime}_{n},$ admit the Knuth relations (KNT) and thus the congruence $\equiv_{\operatorname{plc}}$ of $\mathcal{P}_{n}$ implies the equivalence ${\equiv^{\operatorname{co}}_{\operatorname{plc}}}$.

Proof of Theorem 7.2.

Proof by induction on $n$. The case of $n=3$ has been proven in Example A.1. Assuming the implication holds for $n-1$, we prove it by cases for $n$, heavily basing on the Encoding Algorithm  6.19 of configuration tableaux.

For an easy exposition, we write the words $a_{1}^{\prime},\dots,a_{n}^{\prime}$ in $\mathcal{A}_{n}^{*}$, each is of length $(n-1)$, as a table

[TABLE]

where the empty spaces stand for absent letters. To indicate that an $n$-configuration tableau $\mathfrak{C}$ is considered with respect to the sub-alphabet $\{a_{k},\dots,a_{n}\}\subset\mathcal{A}_{n}$, we denote it by $\mathfrak{C}^{(k:n)}$; $\mathfrak{C}^{(n)}_{w}:=\mathfrak{C}^{(1:n)}_{w}$ denotes the $n$-configuration tableau of a word $w\in\mathcal{A}^{*}_{n}$ over the whole alphabet $\mathcal{A}_{n}=\{a_{1},\dots,a_{n}\}$.

As the word $a^{\prime}_{\ell}$ has a decreasing order, the encoding of its $i$’th letter in $\mathfrak{C}^{(n)}$ “bumps” all the pervious letters, resulting in a tableau of the form  (A.17), which is similar to encoding its $(i-1)$-prefix in a configuration tableau of a fewer letters. Thus, we obtain the following observations for the encoding $a^{\prime}_{\ell}\looparrowright\mathfrak{C}$ of $a^{\prime}_{\ell}$ in $\mathfrak{C}$.

• (A)

If $\ell<n$ then $a^{\prime}_{\ell}=ua_{1}$ and the upper part of $\mathfrak{C}_{ua_{1}}^{(n)}$ is equal to $\mathfrak{C}_{u}^{(2:n)}$.

• (B)

When $\ell=n$, the right part of $\mathfrak{C}_{a^{\prime}_{n}}^{(n)}$ is equal to $\mathfrak{C}_{a^{\prime}_{n}}^{(2:n)}$.

Let $a^{\prime}=a_{p}^{\prime},b^{\prime}=a_{q}^{\prime}$, $c^{\prime}=a_{r}^{\prime}$, where $p\leq q\leq r$. By (KNT) we have to prove the following equalities

$\operatorname{KN1}$:

$a^{\prime}\;c^{\prime}\;b^{\prime}=c^{\prime}\;a^{\prime}\;b^{\prime}\ \text{if }p\leq q<r$,

$\operatorname{KN2}$:

$b^{\prime}\;a^{\prime}\;c^{\prime}=b^{\prime}\;c^{\prime}\;a^{\prime}\ \text{if }p<q\leq r$.

To do so we proceed by cases, depending on the values of $p$, $q$, and $r$.

Case I. $r<n$:  Then also $p,q<n$, and we can write $a^{\prime}_{p}=u_{p}a_{1}$, $a^{\prime}_{q}=u_{q}a_{1}$, and $a^{\prime}_{r}=u_{r}a_{1}$. All have the same terminating letter $a_{1}$ (i.e., the $1$-suffix), and observation (A) holds sequentially for concatenations of $a^{\prime}_{p}$, $a^{\prime}_{q}$, and $a^{\prime}_{r}$, e.g. see (A.8). Therefore

[TABLE]

for $i,j,k\in\{p,q,r\}$. Thus

[TABLE]

and this case is completed by induction.

Case II. $r=n$:  We have several sub-cases, depending on the indices $p,q$, which we recall satisfy $p\leq q\leq n$.

1. (a)

$p\leq q<n-1<r=n$:  Write $a^{\prime}_{p}=u_{p}a_{1}$, $a^{\prime}_{q}=u_{q}a_{1}$, $a^{\prime}_{r}=u_{r}a_{2}$. Then, $a_{2}$ is the $1$-suffix of $a^{\prime}_{r}$, and it is the smallest letter in $a^{\prime}_{r}$, which is also the $1$-suffix of both $u_{p}$ and  $u_{q}$, since $p,q<n-1$. Thus, observation (A) holds inductively, and hence

[TABLE]

This case is then completed by induction as in (A.19). 2. (b)

$p<q=n-1<r=n$:  Write $a^{\prime}_{p}=ua_{2}a_{1}$, $a^{\prime}_{q}=va_{3}a_{1}$, and $a^{\prime}_{r}=va_{3}a_{2}$ (e.g. see (A.4) and (A.12)).

$\operatorname{KN1}$: The first row of $\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}}$ is $110\cdots 0$. The encoding $a^{\prime}_{q}\looparrowright\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}}$ increments $\lambda_{1,3}$ by $a_{3}\looparrowright\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}v}$ to obtain $\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}va_{3}}|_{\operatorname{row}_{1}}=1110\cdots 0$, and then $a_{1}\looparrowright\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}va_{3}}$ gives $\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}a^{\prime}_{q}}|_{\operatorname{row}_{1}}=2010\cdots 0$. For $\mathfrak{C}_{a^{\prime}_{r}a^{\prime}_{p}}$ we have the same $\mathfrak{C}_{a^{\prime}_{r}a^{\prime}_{p}}|_{\operatorname{row}_{1}}=110\cdots 0$, and as before $a^{\prime}_{q}\looparrowright\mathfrak{C}_{a^{\prime}_{r}a^{\prime}_{p}}$ results in $\mathfrak{C}_{a^{\prime}_{r}a^{\prime}_{p}a^{\prime}_{q}}|_{\operatorname{row}_{1}}=2010\cdots 0$. Then, by induction on $a^{\prime\prime}=ua_{2}$, $b^{\prime\prime}=va_{3}$, $c^{\prime\prime}=va_{3}$, we have

[TABLE]

$\operatorname{KN2}$: The first row of $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}}$ is $20\cdots 0$, then $a^{\prime}_{r}\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}}$ increments $\lambda_{1,2}$ to obtain $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}a^{\prime}_{r}}|_{\operatorname{row}_{1}}=210\cdots 0$. For $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}}$ we have $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}}|_{\operatorname{row}_{1}}=110\cdots 0$, and $a^{\prime}_{p}\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}}$ results in $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}a^{\prime}_{p}}|_{\operatorname{row}_{1}}=210\cdots 0$, since it increments $\lambda_{1,2}$ and immediately decrements it by the encoding of $a_{1}$, which increments $\lambda_{1,1}$. Then, by induction on $a^{\prime\prime}=ua_{2}$, $b^{\prime\prime}=va_{3}$, $c^{\prime\prime}=va_{3}$, we have

[TABLE] 3. (c)

$p=q=n-1<r=n$:  Write $a^{\prime}_{p}=a^{\prime}_{q}=ua_{3}a_{1}$ and $a^{\prime}_{r}=va_{3}a_{2}$ (e.g. (A.8)). The first row of $\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}}$ is $110\cdots 0$, then $a^{\prime}_{q}\looparrowright\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}}$ increments $\lambda_{1,3}$ and $\lambda_{1,1}$, and increments $\lambda_{1,2}$ to obtain $\mathfrak{C}_{a^{\prime}_{p}a^{\prime}_{r}a^{\prime}_{q}}|_{\operatorname{row}_{1}}=2010\cdots 0$. On the other hand $\mathfrak{C}_{a^{\prime}_{r}a^{\prime}_{p}}|_{\operatorname{row}_{1}}=101\cdots 0$, and $a^{\prime}_{q}\looparrowright\mathfrak{C}_{a^{\prime}_{r}a^{\prime}_{p}}$ gives $\mathfrak{C}_{a^{\prime}_{r}a^{\prime}_{p}a^{\prime}_{q}}|_{\operatorname{row}_{1}}=2010\cdots 0$. But $u=v$, and thus

[TABLE]

(In this case, $\operatorname{KN2}$ is not relevant here.) 4. (d)

$p<n-1<q=r=n$:  Let $a^{\prime}_{p}=ua_{2}a_{1}$ and $a^{\prime}_{q}=a^{\prime}_{r}=va_{3}a_{2}$. The first row of $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}}$ is $110\cdots 0$, then $a^{\prime}_{r}\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}}$ increments $\lambda_{1,2}$ to obtain $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}a^{\prime}_{r}}|_{\operatorname{row}_{1}}=120\cdots 0$. On the other hand $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}}|_{\operatorname{row}_{1}}=020\cdots 0$, and $a^{\prime}_{p}\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}}$ gives $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}a^{\prime}_{p}}|_{\operatorname{row}_{1}}=120\cdots 0$. Thus

[TABLE]

by induction on $a^{\prime\prime}=ua_{2}$, $b^{\prime\prime}=c^{\prime\prime}=va_{3}$. (In this case, $\operatorname{KN1}$ is not relevant here.) 5. (e)

$p=n-1<q=r=n,$:  Write $a^{\prime}_{p}=ua_{3}a_{1}$ and $a^{\prime}_{q}=a^{\prime}_{r}=va_{3}a_{2}$ (e.g. see (A.16)). The first row of $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}}$ is $1010\cdots 0$, then $v\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}}$ fills the forth column by $1$, which is bumped by $a_{3}\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}v}$. So $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}va_{3}}|_{\operatorname{row}_{1}}=1020\cdots 0$, and $a_{2}\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}va_{3}}$ gives $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{p}a^{\prime}_{r}}|_{\operatorname{row}_{1}}=1110\cdots 0$, as $\lambda_{1,3}$ is decremented by encoding $a_{2}$. On the other hand $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}}|_{\operatorname{row}_{1}}=020\cdots 0$, and $a^{\prime}_{p}\looparrowright\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}}$ gives $\mathfrak{C}_{a^{\prime}_{q}a^{\prime}_{r}a^{\prime}_{p}}|_{\operatorname{row}_{1}}=1110\cdots 0$. Thus

[TABLE]

since $u=v$. (In this case, $\operatorname{KN1}$ is not relevant here.)

The above cases show that involving a new letter reflects inductively on extending the tableaux by a bottom row, and the proof is completed. ∎