Bender-Knuth involutions on linear extensions of posets

TL;DR

This paper investigates the structure and properties of the permutation group generated by Bender-Knuth moves on linear extensions of posets, analyzing group relations and conditions for equality with the symmetric group.

Contribution

It introduces the group $\

Findings

Identifies posets where Bender-Knuth moves satisfy cactus relations.

Characterizes when $\

concludes with criteria for $\

Abstract

We study the permutation group $\mathcal{BK}_P$ generated by Bender-Knuth moves on linear extensions of a poset $P$, an analog of the Berenstein-Kirillov group on column-strict tableaux. We explore the group relations, with an emphasis on identifying posets $P$ for which the cactus relations hold in $\mathcal{BK}_P$. We also examine $\mathcal{BK}_P$ as a subgroup of the symmetric group $\mathfrak{S}_{\mathcal{L}(P)}$ on the set of linear extensions of $P$ with the focus on analyzing posets $P$ for which $\mathcal{BK}_P = \mathfrak{S}_{\mathcal{L}(P)}$.

Equations

$$(3,2)=\text{ydiagram}3,2.$$

$${\operatorname{CST}}((3,2),(1,2,1,1))=\left\{\footnotesize\;\ytableau 1&22\\ 34\none\;,\;\ytableau 1&23\\ 24\none\;,\;\ytableau 1&24\\ 23\none\;\right\}.$$

$${\operatorname{SYT}}((3,2))=\left\{\footnotesize\;\ytableau 1&23\\ 45\none\;,\;\ytableau 1&24\\ 35\none\;,\;\ytableau 1&25\\ 34\none\;,\;\ytableau 1&34\\ 25\none\;,\;\ytableau 1&35\\ 24\none\;\right\}.$$

$$T=\;\ytableau 1&11122223\\ 223334\\ 3445\;.$$

$$S=\;\ytableau\none&\none\none\none 2*(yellow)2*(yellow)2*(yellow)2*(pink)3\\ 2*(yellow)2*(pink)3*(pink)33\\ 3\;.$$

$$S^{\prime}=\;\ytableau\none&\none\none\none 2*(pink)2*(yellow)3*(yellow)3*(yellow)3\\ 2*(pink)2*(pink)2*(yellow)33\\ 3\;,$$

$$t_{2}(T)=\;\ytableau 1&11122333\\ 222334\\ 3445\;.$$

$$D_{\lambda }=\{(i,j):1\le j\le {\lambda }_i\}.$$

$$\;\ytableau 1&246\\ 35\\ 7\;\quad\longleftrightarrow\quad\;\vbox{\hbox{ \leavevmode\hbox to82.3pt{\vbox to59.15pt{\pgfpicture\makeatletter\hbox{\hskip 48.26215pt\lower-58.60551pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\centering{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0}\pgfsys@invoke{ 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$$(p_1,\dots ,p_{i+1},p_i,\dots ,p_n)$$

$$I:=\{p\in P\setminus Q:\text{there exists }q\in Q\text{ with }q>p\},\text{ and }F:=P\setminus (Q\bigsqcup I).$$

$$T:\mathcal{L}(P+Q)$$

$$\text{by}\ell$$

$$T(q_{jk}\ell )=(q_{m-n,m}({\ell }_P),q_{k-m,k-j-n-1}({\ell }_Q),(q_{jk}S)(P),(q_{jk}S)(Q)).$$

$$T(q_{jk}\ell )=(q_{m-n,m}({\ell }_P),q_{k-m,k-j-n-1}({\ell }_Q),(q_{jk}S)(P),(q_{jk}S)(Q))$$

$$(m-n)-i^{\prime }$$

$$=|S(P)\cap [i+1,j-1]|+1\ge 2$$

$$q_{jk}=q_{k-1}{q}_{k-j}{q}_{k-1}=q_{k-1}{q}_1{q}_{k-1}=q_{k-1}{t}_1{q}_{k-1}=q_{k-1}^2=1,$$

$$\begin{array}[]{r l}q_{j}&=t_{1}(t_{2}t_{1})\ldots(t_{j}t_{j-1}\ldots t_{1})=t_{1}(t_{1})\ldots(t_{j}\ldots t_{3}t_{1})\\[5.0pt] &=t_{1}^{j}(t_{3})(t_{4}t_{3})\ldots(t_{j}t_{j-1}\ldots t_{3})=t_{1}^{j}q^{\prime}_{j}\end{array}$$

$$q_j^{\prime }=(t_2)(t_3{t}_2)(t_4{t}_3{t}_2)\dots (t_j{t}_{j-1}\dots t_3{t}_2)=(t_3)(t_4{t}_3)\dots (t_j{t}_{j-1}\dots t_3).$$

$$\begin{array}[]{rl}q_{jk}&=q_{k-1}q_{k-j}q_{k-1}=(t_{1}^{k-1}q^{\prime}_{k-1})(t_{1}^{k-j}q^{\prime}_{k-j})(t_{1}^{k-1}q^{\prime}_{k-j})\\[5.0pt] &=t_{1}^{k-j}(q^{\prime}_{k-1}q^{\prime}_{k-j}q^{\prime}_{k-1})=:t_{1}^{k-j}q^{\prime}_{jk}.\end{array}$$

$$S=\;\ytableau 1&234\\ \none 567\\ \none\none 89\;\quad\Rightarrow\quad T=\;\ytableau*(yellow)1&1234\\ *(yellow)2*(yellow)5567\\ *(yellow)3*(yellow)6*(yellow)889\\ *(yellow)4*(yellow)7*(yellow)9\none\none\;.$$

$$\lambda :\mathcal{L}(P\oplus Q)$$

$$\ell$$

$$\psi :{\mathcal{B}\mathcal{K}}_{P\oplus Q}$$

$$t_i$$

$$t_i^{\prime }=\{\begin{array}{ll}(t_i,1)& \text{if}i\le |P|-1\\ (1,1)& \text{if}i=|P|\\ (1,t_{i-|P|})& \text{if}i\ge |P|+1.\end{array}$$

$$n=\sum _{j=1}^r{n}_j.$$

$$|\mathcal{L}(P)|\cdot |\mathrm{Stab}(\ell )|$$

$$\frac{n!}{n_1!\cdots n_r!}\cdot |\mathrm{Stab}(\ell )|$$

$$|\mathrm{Stab}(\ell )|$$

$$v_1<v_2<\cdots <v_{a+1}>v_{a+2}>\cdots >v_{a+b+1}<v_{a+b+2}<\cdots <v_{a+b+c+1}.$$

$$v_1$$

Full text

1. 1 Introduction

2. 1.1 Outline of the paper

3. 1.2 Acknowledgments

4. 2 Basic definitions and constructions

5. 2.1 Column-strict tableaux and Bender–Knuth moves

6. 2.2 Bender–Knuth moves on linear extensions

7. 2.3 Poset operations

8. 3 Relations in $\mathcal{BK}_{P}$

9. 3.1 Relations in $\mathcal{BK}_{P}$ and convex induced subposets

10. 3.2 LE-cactus posets

11. 3.2.1 Disjoint unions of LE-cactus posets

12. 3.2.2 Ordinal sums of LE-cactus posets

13. 3.3 Minuscule, d-complete, and jeu-de-taquin posets

14. 4 LE-symmetric posets

15. 4.1 Duals and ordinal sums

16. 4.2 LE-primitive posets and disjoint unions

17. 4.3 LE-symmetric families

18. 4.3.1 Conjectural LE-symmetric families and computational results

19. 5 Size of BK groups

20. 5.1 Duals, ordinal sums, and disjoint unions

21. 5.2 Stabilizer size of linear extensions under BK action

1. Introduction

First introduced by Bender and Knuth in their study of enumerations of plane partitions and Schur polynomials [BK72], the Bender–Knuth (BK) moves, a certain family of involutions on the set of column-strict (semi-standard) Young tableaux, have seen a wide range of applications across different areas of combinatorics. They were shown to be equivalent to tableau switching, an involution on pairs of column-strict tableaux, on horizontal border strips of two adjacent letters, together with a swap of these labels [BSS96, PV10]. Berenstein and Kirillov studied an extension of these involutions, called piecewise-linear BK moves, acting on Gelfand–Tsetlin patterns and the relations they satisfy [BK95]. Other interpretations and applications of BK moves appeared in the context of crystals for finite-dimensional complex reductive Lie algebras (e.g., [Hal20]), shifted tableaux (e.g., [Ste90, Rod21]), and Grothendieck polynomials (e.g., [IS14, GGL16]).

Informally, the BK moves $t_{i}$ act on a column-strict tableau by fixing an $i$ (resp. $i+1$) when there is an $i+1$ below (resp. $i$ above), and swapping the contents of the remaining letters $i$ and $i+1$ in each row. The Berenstein–Kirillov (BK) group $\mathcal{BK}$ is defined to be the free group generated by variables $t_{i}$ for all $i\in\mathbb{Z}_{>0}$, modulo the relations satisfied by the BK moves $t_{i}$ when acting on all column-strict tableaux. This group was formally introduced by Berenstein and Kirillov [BK95], who also gave an incomplete list of relations in this group. More recently, Chmutov, Glick, and Pylyavskyy [CGP20] related the BK moves to the cactus group $\mathcal{C}_{n}$, the fundamental group of the moduli space of marked real genus zero stable curves (see, e.g., [Dev99, DJS03, HK06, Hal20]). In [CGP20], they showed that the BK moves $t_{i}$ acting on tableaux satisfy the relations for the usual generators of $\mathcal{C}_{n}$, giving a group homomorphism from $\mathcal{C}_{n}$ to the subgroup $\mathcal{BK}_{n}$ of the BK group generated by $\{t_{1},\ldots,t_{n-1}\}$, and yielding new relations previously unknown in $\mathcal{BK}$. Their result is one of the motivations of this work.

Recall that a linear extension of a poset $P$ is a linear order that is compatible with $P$. Haiman [Hai92] and Malvenuto and Reutenauer [MR94] introduced an analog of the BK moves $t_{i}$ on linear extensions of a poset $P$, which swap two adjacent letters $i$ and $i+1$ when they label incomparable elements of $P$ and fix them otherwise, and used them to study promotion and evacuation, operators on linear extensions first defined by Schützenberger [Sch76, Sch77]. A survey on basic properties and generalizations of these operators can be found in [Sta09].

An important tool to study BK moves on linear extensions is the linear extension graph of a poset $P$, the graph whose vertices are labelled by linear extensions of $P$ and edges are given by the BK moves that swap corresponding linear extensions. Linear extension graphs were first introduced by Pruesse and Ruskey [PR91], and previously used in the study of linear extension generation (see, e.g., [Rus92, Sta92, Wes93, Naa00, BM13]) as well as Markov chains on the set of linear extensions (e.g., [AKS14]). A survey on linear extension graphs can be found in [Mas10].

In this context, we may define an analog of the Berenstein–Kirillov group on linear extensions of a given poset $P$, denoted by $\mathcal{BK}_{P}$. The goal of this paper is to study properties of this group $\mathcal{BK}_{P}$, attempting to characterize the classes of posets $P$ for which $\mathcal{BK}_{P}$ enjoys various properties.

1.1. Outline of the paper

Section 2 provides basic definitions and constructions that are fundamental to this paper, including column-strict tableaux, linear extensions of posets, and BK moves on various combinatorial objects. Given a poset $P$, the Berenstein–Kirillov group of $P$, $\mathcal{BK}_{P}$ is defined to be the permutation group generated by BK moves $t_{i}$ on linear extensions of $P$.

In Section 3, we focus on the relations in $\mathcal{BK}_{P}$. In particular, we will identify the posets $P$ for which the trivialization relations (Proposition 3.1) and braid relations (Proposition 3.8) hold; note that the latter was previously observed by [AKS14]. Progress is made towards understanding the LE-cactus posets, that is, the posets for which the cactus relations hold in $\mathcal{BK}_{P}$.

Finally, Sections 4 and 5 are dedicated to the study of the group $\mathcal{BK}_{P}$ as a permutation group on the set ${\mathcal{L}}(P)$ of all linear extensions of $P$. Section 4 focuses on understanding posets $P$, called LE-symmetric posets, for which $\mathcal{BK}_{P}$ equals the full symmetric group on ${\mathcal{L}}(P)$. More specifically, we classify all disconnected LE-symmetric posets (Theorem 4.12) and LE-symmetric series-parallel posets (Corollary 4.13), and exhibit a few other families of connected LE-symmetric posets. Section 5 explores the cardinality of $\mathcal{BK}_{P}$.

1.2. Acknowledgments

We are indebted to Vic Reiner for initiating the subject of study and providing extremely valuable guidance. We would like to thank Gregg Musiker and the 2022 University of Minnesota Combinatorics and Algebra REU staff for organizing the program. Finally, we thank Pasha Pylyavskyy, Joel Kamnitzer, Iva Halacheva, and Sylvester Zhang for helpful discussions. This research is partially supported by RTG grant NSF/DMS-1745638.

2. Basic definitions and constructions

In this section, we will provide the background on several algebraic and combinatorial objects that are fundamental to this paper, including column-strict and standard Young tableaux, linear extensions, Bender–Knuth involutions, and poset operations.

2.1. Column-strict tableaux and Bender–Knuth moves

Recall that a partition $\lambda$ of a positive integer $N$ is a tuple of integers $\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n})$ such that $\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{n}\geq 0$ and $\sum_{i=1}^{n}\lambda_{i}=N$. The Young diagram of shape $\lambda$ is a finite collection of boxes arranged in left-justified rows, where the $i^{\mathrm{th}}$ row has $\lambda_{i}$ boxes. A column-strict (semi-standard) tableau $T$ of shape $\lambda$ is a filling of the Young diagram of $\lambda$ that is weakly increasing along rows and strictly increasing along columns. A tableau $T$ is said to have content $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ if there are $\alpha_{i}$ occurrences of $i$ in $T$ for each $i=1,\ldots,n$; when the content is $\alpha=(1,1,\ldots,1)$, the tableau $T$ is called a standard Young tableau. Denote the set of all column-strict tableaux of shape $\lambda$ and content $\alpha$ by ${\operatorname{CST}}(\lambda,\alpha)$, and the set of standard Young tableaux of shape $\lambda$ by ${\operatorname{SYT}}(\lambda)$.

Example 2.1*.*

The Young diagram of shape $(3,2)$ is

[TABLE]

The collection of column-strict tableaux of shape $(3,2)$ and content $(1,2,1,1)$ is

[TABLE]

Meanwhile, that of standard Young tableaux of the same shape is

[TABLE]

The primary combinatorial object of interest in this paper is the Bender–Knuth moves (or BK moves for short), which were originally defined on column-strict tableaux by [BK72] and further studied by several authors, e.g., [Sch72, BZ90, BK95, CGP20]. Fix a partition $\lambda$, and let $T$ be a column-strict tableau of shape $\lambda$ and content $\alpha=(\alpha_{1},\ldots,\alpha_{n})$.

Definition 2.2**.**

The Bender–Knuth move $t_{i}$ for $1\leq i\leq n-1$ is an involution (i.e., a permutation of order $2$) of the set of column-strict tableaux of shape $\lambda$ that sends $T$ to the tableau obtained from the following procedure:

1. (1)

Let $S$ be the skew tableau obtained by taking only the boxes of $T$ with entry equal $i$ and $i+1$; 2. (2)

Observe that each row of $S$ contains

1. (a)

$a$ entries equal to $i$ that lie directly above an $i+1$, 2. (b)

$b$ entries equal to $i$ that are alone in their columns, 3. (c)

$c$ entries equal to $i+1$ that are alone in their columns, and 4. (d)

$d$ entries equal to $i+1$ that lie directly below an $i$

for some $a,b,c,d\geq 0$; 3. (3)

Construct a skew tableau $S^{\prime}$ by swapping $b$ and $c$ in each row of $S$; 4. (4)

Define $t_{i}(T)$ to be the tableau obtained by replacing $S$ with $S^{\prime}$ in $T$.

Example 2.3*.*

Consider the action of the BK move $t_{2}$ on the tableau

[TABLE]

The skew tableau $S$ containing only boxes of entry $2$ and $3$ in $T$ is

[TABLE]

Following Definition 2.2, observe that in the first row $b=3$ (yellow boxes) and $c=1$ (pink boxes), so after swapping we get $b^{\prime}=1$ and $c^{\prime}=3$. The second row has $b=1$ and $c=2$, so we turn a $3$ into a $2$, whereas the third row has $b=c=0$ and hence is fixed. The new skew tableau $S^{\prime}$ obtained after this process has the form

[TABLE]

and hence

[TABLE]

Let $\mathcal{BK}$, called the Berenstein–Kirillov (BK) group, denote the free group generated by the variables $t_{i}$ $\left(i\in\mathbb{Z}_{>0}\right)$ modulo the relations satisfied by the BK moves $t_{i}$ when acting on all column-strict tableaux of all possible shapes. The relations in this group were first studied by Berenstein and Kirillov [BK95], who gave the following few:

1. (1)

$t_{i}^{2}=1$ for all $i\geq 1$; 2. (2)

$(t_{i}t_{j})^{2}=1$ for all $i,j\geq 1$ with $|i-j|\geq 2$; 3. (3)

$(t_{1}q_{i})^{4}=1$ where $q_{i}=t_{1}(t_{2}t_{1})\ldots(t_{i}t_{i-1}\ldots t_{1})$ and $i\geq 3$; and 4. (4)

$(t_{1}t_{2})^{6}=1$.

More recently, Chmutov, Glick, and Pylyavskyy found another relation in this group generalizing relation (3):

1. (5)

$(t_{i}q_{jk})^{2}=1$ for $q_{jk}=q_{k-1}q_{k-j}q_{k-1}$ whenever $i+1<j<k$,

which interestingly gives a group homomorphism from the cactus group $\mathcal{C}_{n}$ (see, e.g., [Dev99, HK06, Hal20]) to the subgroup $\mathcal{BK}_{n}$ of the BK group generated by $\{t_{1},\ldots,t_{n-1}\}$ [CGP20]. For this reason, we call relation (5) the cactus relation. Note that $q_{i}$ consists of an iterated product of $\partial_{j}=t_{j}\dots t_{2}t_{1}$; both of these have been previously studied as operators on column-strict tableaux, called evacuation and promotion111More precisely, the operators $\partial_{i}$ and $q_{i}$ as defined here are partial promotion and evacuation, as they act only on the subtableaux containing the entries $1,2,\dots,i+1$, instead of the full tableaux. respectively, notably by [Sch72, Sch76, Sch77, BK95]. It is known that evacuation $q_{i}$ is an involution, and so is $q_{jk}$. Thus the cactus relations are equivalent to commutativity of $t_{i}$ and $q_{jk}$ for all $i+1<j<k$.

2.2. Bender–Knuth moves on linear extensions

Let $(P,\leq_{P})$ be an $n$-element poset.

Definition 2.4**.**

A linear extension of $P$ is a linear order $(\ell,\leq_{\ell})$ which extends $P$ in the sense that $x\leq_{P}y$ implies $x\leq_{\ell}y$.

More precisely, a linear extension $\ell$ of $P$ is a bijection $\ell:P\to\{1,\dots,n\}=:[n]$ such that $x<_{P}y$ implies $\ell(x)<\ell(y)$. Alternatively, a linear extension $\ell$ of $P$ can be interpreted as an ordered list $\ell=(p_{1},\ldots,p_{n})$ of elements of $P$ such that $p_{i}<p_{j}$ implies $i<j$; the index of an element $p$ on this list is precisely its label $\ell(p)$. The set of all linear extensions of $P$ is denoted by ${\mathcal{L}}(P)$. The example below shows the poset $P$ defined by the relations $a,b<c,d$ (left) and a linear extension $\ell$ of $P$ (right). Observe that the labelling system on $\ell$ satisfies Definition 2.4. In the alternative notation, we have $\ell=(b,a,d,c)$.

a$$b$$c$$d$$2$$1$$4$$3

Given a partition $\lambda=(\lambda_{1},\dots,\lambda_{n})$, the Young diagram of $\lambda$ is a subset $D_{\lambda}$ of $\{1,2,\ldots\}\times\{1,2,\ldots\}$ defined by

[TABLE]

The Ferrers poset $F_{\lambda}$ of $\lambda$ is defined to be the set $D_{\lambda}$ with a partial order generated by the covering relations $(i,j)<(i,j+1)$ and $(i,j)<(i+1,j)$. It is well-known that there is a one-to-one correspondence between the standard Young tableaux of shape $\lambda$ and the linear extensions of the corresponding Ferrers poset $F_{\lambda}$. The example below shows a standard Young tableau of shape $\lambda=(4,2,1)$ and its corresponding linear extension of the Ferrers poset $F_{\lambda}$.

[TABLE]

Observe that if $T$ is a standard Young tableau, by Definition 2.2, the BK move $t_{i}$ acts on $T$ by simply switching $i$ and $i+1$ if they label non-adjacent boxes and fixing $T$ otherwise. Via the above bijection, there is an analog of the BK moves on linear extensions of Ferrers posets, which naturally extends to any arbitrary poset. This generalization was first introduced by Haiman [Hai92] as well as Malvenuto and Reutenauer [MR94] to study promotion and evacuation on linear extensions, which are analogs of operators on column-strict tableaux previously studied by Schützenberger [Sch72, Sch76, Sch77].

Definition 2.5**.**

For a given $n$-element poset $P$ and $1\leq i\leq n-1$, the Bender–Knuth move $t_{i}$ is an involution of ${\mathcal{L}}(P)$ that sends $\ell=(p_{1},\ldots,p_{n})$ to

[TABLE]

if $p_{i}$ and $p_{i+1}$ are incomparable, and fixes $\ell$ otherwise.

As in the case with column-strict tableaux, two important operators generated by the BK moves are (partial) promotion $\partial_{i}=t_{i}t_{i-1}\ldots t_{1}$ (by convention, $\partial_{0}=1$) and (partial) evacuation $q_{i}=\partial_{0}\partial_{1}\dots\partial_{i}$. We refer the reader to [Sta09] for a careful analysis of these operators; note that the inclusion of the identity operator $\partial_{0}$ in $q_{i}$ is motivated by a procedural description in the reference of the evacuation $q_{i}$ as a series of $i+1$ partial promotions. For the purpose of this paper, we emphasize a useful procedural description of the promotion operator $\partial_{i}$:

Definition 2.6**.**

Given an $n$-element poset $P$ and $1\leq i\leq n-1$, the promotion operator $\partial_{i}$ is a permutation of ${\mathcal{L}}(P)$ that sends $\ell\in{\mathcal{L}}(P)$ to the linear extension of $P$ obtained from the following procedure:

1. (1)

Consider the subposet formed by the elements $\ell^{-1}(1),\ell^{-1}(2),\ldots,\ell^{-1}(i+1)$; 2. (2)

Let $p_{1}=\ell^{-1}(1)$. Remove the label $1$ from $p_{1}$; 3. (3)

Among the elements covering $p_{1}$, let $p_{2}$ be the element with the smallest label $\ell(p_{2})$; 4. (4)

Slide this label down to $p_{1}$; 5. (5)

Repeat until we reach a maximal element $p_{k}$. Label $p_{k}$ with $i+2$ and subtract all labels on the entire subposet by $1$.

We call $p_{1}<p_{2}<\ldots<p_{k}$ a promotion chain. An immediate consequence of this interpretation is that when a promotion operator $\partial_{i}$ acts on a linear extension of a disjoint union of posets $P_{1},\ldots,P_{n}$, the only component $P_{i}$ whose labels’ relative order may be affected is the one containing $\ell^{-1}(1)$.

Given a finite poset $P$, its linear extension graph is the graph whose vertices are labelled by linear extensions of $P$ and edges are given by the BK moves that swap corresponding linear extensions (e.g., see Figure 1). Linear extension graphs were first introduced by Pruesse and Ruskey [PR91], and previously used in the study of linear extension generation (e.g., [Rus92, Sta92, Wes93, Naa00, BM13]) and Markov chains on ${\mathcal{L}}(P)$ (e.g., [AKS14]).

We may define an analog of the BK group in this context.

Definition 2.7**.**

For a given $n$-element poset $P$, the Berenstein–Kirillov group of $P$, denoted by $\mathcal{BK}_{P}$, is the permutation group of the set ${\mathcal{L}}(P)$ generated by the BK moves $t_{i}$ ($1\leq i\leq n-1$).

While never explicitly studied, some relations are known to hold in $\mathcal{BK}_{P}$ for all posets $P$, e.g.,

1. (1)

$t_{i}^{2}=1$ for all $i\geq 1$; 2. (2)

$(t_{i}t_{j})^{2}=1$ for all $i,j\geq 1$ with $|i-j|\geq 2$; and 3. (3)

$(t_{i}t_{i+1})^{6}=1$ for all $i\geq 1$ [Sta09].

When $P$ is a Ferrers poset, the result of [CGP20] implies that the cactus relations hold in $\mathcal{BK}_{P}$, alongside the above. Outside of these few exceptions, very little is known about the group relations in $\mathcal{BK}_{P}$.

The following statement about $\mathcal{BK}_{P}$ as a permutation group of ${\mathcal{L}}(P)$ has been widely utilized in different forms by, e.g., [Pak02, LW08, DMR16, DK21]. A proof was explicitly given by Ayyer, Klee, and Schilling (see Proposition 4.1 of [AKS14]), by showing that linear extension graphs are strongly connected.

Proposition 2.8**.**

$\mathcal{BK}_{P}$* is a transitive subgroup of ${\mathfrak{S}}_{{\mathcal{L}}(P)}$.*

Similar to the relations in $\mathcal{BK}_{P}$, very little has been established about its relative size as a subgroup of the symmetric group on ${\mathcal{L}}(P)$. For instance, Vershik and Tsilevich studied the BK groups of Ferrers posets in the form of permutation groups on Young graphs [VT19]. The rest of the paper will be guided by the following motivating questions about properties of $\mathcal{BK}_{P}$:

Question 2.9*.*

What further relations hold in $\mathcal{BK}_{P}$? In particular, for which posets $P$ do the cactus relations hold in $\mathcal{BK}_{P}$?

Question 2.10*.*

For which posets $P$ does $\mathcal{BK}_{P}={\mathfrak{S}}_{{\mathcal{L}}(P)}$? More generally, what can we say about the relative size of $\mathcal{BK}_{P}$ as a transitive subgroup of ${\mathfrak{S}}_{{\mathcal{L}}(P)}$?

As mentioned in the introduction, Question 2.9 will be addressed in Section 3, whereas Question 2.10 will be investigated in Sections 4 and  5.

2.3. Poset operations

We fix the notations for various poset operations discussed in this paper. Let $P$ and $Q$ be finite posets. Let $P\oplus Q$ denote the ordinal sum of $P$ and $Q$, where each element of $P$ is less than every element of $Q$. Note that the ordinal sum operation is not commutative; in general, $P\oplus Q\neq Q\oplus P$. Let $P+Q$ denote the disjoint union of $P$ and $Q$. On the contrary, this operation is commutative. Finally, let $P^{*}$ be the dual of the poset $P$, defined by inverting the order of $P$ in the sense that $x\leq_{P^{*}}y$ if and only if $y\leq_{P}x$.

3. Relations in $\mathcal{BK}_{P}$

In this section, we discuss the group relations in $\mathcal{BK}_{P}$, including the trivialization, braid, and cactus relations. The most important findings are located in Section 3.2, where we study LE-cactus posets–posets for which the cactus relations hold in their BK groups–and their behaviors under disjoint union and ordinal sum operations, and Section 3.3, where we study several potential families of such posets.

3.1. Relations in $\mathcal{BK}_{P}$ and convex induced subposets

Let $P$ be an $n$-element poset. Recall that a relation in $\mathcal{BK}_{P}$ is an equation $w=1$, where $w$ is a word in the alphabet $\{t_{1},\dots,t_{n-1}\}$. The first natural family of relations in $\mathcal{BK}_{P}$ that we will discuss is the relations $t_{i}=1$, called the trivialization of $t_{i}$.

Proposition 3.1**.**

For $1\leq i\leq n-1$, the relation $t_{i}=1$ holds in $\mathcal{BK}_{P}$ if and only if $P$ can be written as an ordinal sum $P=P_{1}\oplus P_{2}$ where $|P_{1}|=i$.

Proof.

When $P=P_{1}\oplus P_{2}$ with $|P_{1}|=i$, every linear extension $\ell$ of $P$ has $\ell(P_{1})=\{1,2,\ldots,i\}$ and $\ell(P_{2})=\{i+1,i+2,\ldots,n\}$, so $\ell^{-1}(i)<_{P}\ell^{-1}(i+1)$, and $t_{i}$ fixes $\ell$.

Conversely, if $P\neq P_{1}\oplus P_{2}$ whenever $|P_{1}|=i$, we claim that for any order ideal (lower set) $I$ of $P$ with cardinality $|I|=i$, there exist a maximal element $u$ of $I$ and a minimal element $v$ of the upper set $P\setminus I$ such that $u$ and $v$ are incomparable. Suppose not, then for every maximal element $u$ of $I$ and every minimal element $v$ of $P\setminus I$, we have $u<_{P}v$, as $u>_{P}v\notin I$ is not allowed given that $I$ is an order ideal. Therefore, $P=I\oplus(P\setminus I)$, a contradiction.

Let $I$ be an order ideal of $P$ with cardinality $|I|=i$; for example, choose the inverse image $I=f^{-1}(\{1,2,\ldots,i\})$ for any $f\in{\mathcal{L}}(P)$. We now construct a linear extension $\ell$ of $P$ such that $t_{i}(\ell)\neq\ell$. By the previous claim, we may find an incomparable pair of elements $\{u,v\}$ of $P$ where $u$ is maximal in $I$ and $v$ is minimal in $P\setminus I$. Construct a linear extension $\ell$ by first labeling $\ell(u)=i$ and $\ell(v)=i+1$, then using arbitrary linear extensions to label $I\setminus\{u\}$ with $\{1,2,\ldots,i-1\}$ and $(P\setminus I)\setminus\{v\}$ with $\{i+2,i+3,\ldots,n\}$. It is easy to check that $t_{i}$ swaps the labels of $u$ and $v$ in $\ell$, and hence $t_{i}(\ell)\neq\ell$. ∎

Definition 3.2**.**

A poset is indecomposable if it is not an ordinal sum of two or more non-empty posets.

Corollary 3.3**.**

Let $|P|=n$. If $P$ is indecomposable, then $\{t_{1},\dots,t_{n-1}\}$ is an inclusion-minimal generating set in $\mathcal{BK}_{P}$.

Proof.

By Proposition 3.1, there is some linear extension $\ell$ on which $t_{i}$ is not in the stabilizer. Let $|P|=n$. Suppose for contradiction that $w\ell=t_{i}\ell$ for some $w\in\langle t_{1},t_{2},\dots,\hat{t_{i}},\dots,t_{n-1}\rangle$. Then consider the order ideal $I$ such that $\ell(I)=[i]$, i.e., the elements of $P$ with labels $\{1,2,\dots,i\}$. Then any action that is not $t_{i}$ preserves this image $\ell(I)$, so $i+1\notin w\ell(I)$. However, we have $i+1\in t_{i}\ell(I)$, a contradiction. ∎

Next, we present a useful tool for studying the relations in $\mathcal{BK}_{P}$ by examining the convex induced subposets of $P$. Recall that an induced subposet $Q\subseteq P$ is a subset of vertices in $P$ such that for any $x,y\in Q$, $x\leq_{Q}y$ if and only if $x\leq_{P}y$. A convex induced subposet is an induced subposet such that if $x,z\in Q$ and $y\in P$ satisfy $x\leq_{P}y\leq_{P}z$, then $y\in Q$.

Definition 3.4**.**

A relation type in $\mathcal{BK}_{P}$ is a set of relations $w_{i}=1$, where $w_{i}$ is obtained from a fixed word $w$ in $\{t_{1},\dots,t_{n-1}\}$ by translating the indices of all generators $t_{j}$ in $w$ by an integer $i$, such that $1\leq i+j\leq n-1$ for all $j$.

For example, $(t_{1}t_{2})^{6}=1$ is a relation, whereas the relations $(t_{i}t_{i+1})^{6}=1$ form a relation type. The following proposition gives a means to generate relation types in the BK groups for convex induced subposets.

Proposition 3.5**.**

If a relation type holds in $\mathcal{BK}_{P}$, then it also holds in $\mathcal{BK}_{Q}$ for every convex induced subposet $Q$ of $P$.

The proof of this statement makes use of the following lemma.

Lemma 3.6**.**

Let $Q$ be a convex induced subposet of $P$ and $\ell_{1}\in{\mathcal{L}}(Q)$. Then there is a linear extension $\ell_{2}\in{\mathcal{L}}(P)$ such that for some $i\in\mathbb{Z}_{\geq 0}$, $\ell_{2}(v)=\ell_{1}(v)+i$ for all $v\in Q$.

Proof.

Disjointly decompose the poset $P$ into three subposets $I$, $Q$, and $F$, where

[TABLE]

Then the subposet $Q\sqcup I$ of $P$ containing the elements of $Q$ and $I$ will be an order ideal of $P$, and because $Q$ is convex, $I$ is an order ideal of $Q\sqcup I$. Letting $i:=|I|,q:=|Q|,$ and $f:=|F|$, one can therefore form a linear extension $\ell_{2}$ of $P$ having $\ell_{2}^{-1}(\{1,2,\ldots,i\})$ agree with any linear extension of $I$, having $\ell_{2}^{-1}(\{i+1,i+2,\ldots,i+q\})$ agree with $\ell_{1}$ after adding $i$ to the values in $\ell_{1}$, and having $\ell_{2}^{-1}(\{i+q+1,i+q+2,\ldots,i+q+f\})$ agree with any linear extension of $F$ after adding $i+q$ to its values. ∎

Proof of Proposition 3.5.

We prove the contrapositive. Suppose there is a convex induced subposet $Q\subseteq P$ for which the relation type does not hold in $\mathcal{BK}_{Q}$. That is, there exists a relation $w(t_{i},\ldots,t_{i+k})=1$ of this type which fails on a linear extension $\ell_{1}$ of $Q$. By Lemma 3.6, there is a linear extension $\ell_{2}\in{\mathcal{L}}(P)$ such that for some $j\in\mathbb{Z}_{\geq 0}$, $\ell_{2}(v)=\ell_{1}(v)+j$ for all $v\in Q$. It is now easy to check that $w(t_{i+j},\ldots,t_{i+j+k})=1$ fails on $\ell_{2}$. ∎

Lemma 3.6 directly implies an interesting fact about convex induced subposets:

Corollary 3.7**.**

If $Q$ is a convex induced subposet of $P$, then there is an injection $\mathcal{BK}_{Q}\hookrightarrow\mathcal{BK}_{P}$.

As an application of Proposition 3.5, we examine a well-known relation type called the braid relations in $\mathcal{BK}_{P}$ for an $n$-element poset $P$. Recall that the braid relations are of the form $(t_{i}t_{i+1})^{3}=1$ for all $i=1,\ldots,n-2$. The following fact was previously observed by Ayyer, Klee, and Schilling (see Proposition 2.2 of [AKS14]), but with the proof omitted.

Proposition 3.8**.**

The braid relations hold in $\mathcal{BK}_{P}$ if and only if $P$ is a disjoint union of chains.

Proof.

One can check that the only posets of cardinality $3$ that fail to satisfy the relation $(t_{1}t_{2})^{3}=1$ are defined by either $a>b<c$ (V-shaped) or $a<b>c$ (inverted V-shaped). Any poset that contains one of these two posets as an induced subposet contains it as a convex induced subposet, and hence will fail to satisfy $(t_{i}t_{i+1})^{3}=1$ for some $i$ by Proposition 3.5. The posets that do not contain one of these two posets are precisely disjoint unions of chains. Conversely, suppose, by contradiction, that a disjoint union of chains fails to satisfy $(t_{i}t_{i+1})^{3}=1$ for some $i$. Then the induced subposet on the elements with labels $i$, $i+1$, and $i+2$ fails to satisfy $(t_{1}t_{2})^{3}=1$, and hence is one of the two posets mentioned above. But a disjoint union of chains does not have such an induced subposet, so we have a contradiction. ∎

Since the braid relations are the defining relations of the symmetric group ${\mathfrak{S}}_{n}$ and the braid group $B_{n}$, it follows that the natural map (of sets) that sends the generator $\sigma_{i}$ of ${\mathfrak{S}}_{n}$ (resp. $B_{n}$) to $t_{i}$ for all $1\leq i\leq n-1$ gives a well-defined action of ${\mathfrak{S}}_{n}$ (resp. $B_{n}$) on ${\mathcal{L}}(P)$ only when $P$ is a disjoint union of chains.

3.2. LE-cactus posets

Another important family of relations involving the generators $t_{i}$ is the cactus relations, which have the form $(t_{i}q_{jk})^{2}=1$, where $q_{jk}=q_{k-1}q_{k-j}q_{k-1}$ and $q_{j}=t_{1}(t_{2}t_{1})\ldots(t_{j}t_{j-1}\ldots t_{1})$ for any $i+1<j<k$. Recall that when $P$ is a Ferrers poset, the cactus relations hold in the group $\mathcal{BK}_{P}$ for all $i+1<j<k\leq|P|$. One reasonable question is whether the same statement is true for an arbitrary poset. Unfortunately, that is not the case.

Example 3.9*.*

On the $4$-element posets, there is only one eligible cactus relation: $(t_{1}q_{34})^{2}=1$. The only $4$-element connected posets for which this relation fails are the followings:

For a counterexample with $5$ elements, the following poset exhibits an interesting property that all eligible cactus relations fail in its BK group:

The fact that cactus relations do not hold in general on all linear extensions of all posets presents a major deviation from the original BK group $\mathcal{BK}$ on semi-standard Young tableaux, a subset of which–namely the standard Young tableaux–corresponds to linear extensions of Ferrers posets. It remains of interest to examine for which posets $P$ do the cactus relations hold in $\mathcal{BK}_{P}$, i.e., when there is an action of the cactus group $\mathcal{C}_{|P|}$ on ${\mathcal{L}}(P)$.

Definition 3.10**.**

An $n$-element poset $P$ is LE-cactus if the cactus relation $(t_{i}q_{jk})^{2}=1$ holds in the group $\mathcal{BK}_{P}$ for all $2\leq i+1<j<k\leq n$.

We have already encountered one such family of posets.

Theorem 3.11** ([CGP20]).**

Ferrers posets are LE-cactus.

For the rest of this section, we will study properties of LE-cactus posets and identify several sufficient conditions for posets to be LE-cactus. First, observe that Proposition 3.5 does not apply to the cactus relations, since they do not form a relation type: implicitly $q_{jk}$ contains operators $q_{j}=t_{1}(t_{2}t_{1})\ldots(t_{j}t_{j-1}\ldots t_{1})$ which always involve $t_{1}$. However, there is an analogous statement for LE-cactus posets.

Proposition 3.12**.**

A poset $P$ is LE-cactus if and only if every order ideal $I$ of $P$ is LE-cactus.

Proof.

By contrapositive, suppose that there exists an order ideal $I$ of $P$ that is not LE-cactus. Let $|I|=m$, then given any linear extension $\ell_{1}$ of $I$, we can construct a linear extension $\ell$ of $P$ by setting $\ell=\ell_{1}$ on $I$ and $\ell=\ell_{2}+m$ on $P\setminus I$ for some linear extension $\ell_{2}$ of the induced subposet $P\setminus I$. Since $I$ is not LE-cactus, there exists a triple $(i,j,k)$ where $2\leq i+1<j<k\leq m$ and a linear extension $\ell_{1}$ of $I$, such that $(t_{i}q_{jk})^{2}(\ell_{1})\neq\ell_{1}$. Then by the above construction, $(t_{i}q_{jk})^{2}(\ell)\neq\ell$. Thus $P$ is not LE-cactus. The converse immediately follows from the fact that $P$ is an order ideal of itself. ∎

This result is especially useful in eliminating posets that are not LE-cactus by identifying (small) non-LE-cactus ideals sitting in them.

3.2.1. Disjoint unions of LE-cactus posets

We will show that the disjoint union of LE-cactus posets remains LE-cactus. Let $P$ and $Q$ be finite posets. We first define the following map on linear extensions of the disjoint union $P+Q$ to help break down showing commutativity of $t_{i}$ and $q_{jk}$ on an entire linear extension of $P+Q$.

Definition 3.13**.**

Let $P$ and $Q$ be posets with $|P|=m$ and $|Q|=n$. Define the map

[TABLE]

where $S(P)$ and $S(Q)$ are the sets of labels in $\ell$ on $P$ and $Q$, and $\ell_{P}$ and $\ell_{Q}$ are independent linear extensions on $P$ and $Q$ which agree with $\ell$, in the sense that for any $p_{1},p_{2}\in P$, $\ell_{P}(p_{1})\leq\ell_{P}(p_{2})$ if and only if $\ell(p_{1})\leq\ell(p_{2})$, and similarly for $Q$.

It is not difficult to see that $T$ is an injection. It is also straightforward to transfer the action of $t_{i}$ on $\ell$ to an action on the tuple $(\ell_{P},\ell_{Q},S(P),S(Q))$. If $\ell^{-1}(i)$ is in $P$ and $\ell^{-1}(i+1)$ is in $Q$ or vice versa, then $t_{i}$ swaps the label $i$ in $S(P)$ with $i+1$ in $S(Q)$ while keeping $\ell_{P}$ and $\ell_{Q}$ the same. If $i$ and $i+1$ are both in $P$, then $t_{i}$ stabilizes $S(P),S(Q)$, and $\ell_{Q}$, while acting on $\ell_{P}$ by $t_{i^{\prime}}$ where $i^{\prime}=|S(P)\cap[i]|$.

Example 3.14*.*

Consider the following linear extension $\ell$ of a disjoint union $P+Q$ (where the posets $P$ and $Q$ underlie the left and right components, respectively):

1$$4$$2$$6$$5$$9$$3$$7$$8$$10

The map $T$ sends this linear extension to the tuple containing linear extensions $\ell_{P}$ and $\ell_{Q}$:

1$$3$$2$$5$$4$$6$$1$$2$$3$$4$$\ell_{P}$$\ell_{Q}

and the sets of labels $S(P)=\{1,2,4,5,6,9\}$ and $S(Q)=\{3,7,8,10\}$.

Observe that if we act on $\ell$ by $t_{3}$, the resulting $\ell_{P}$ and $\ell_{Q}$ stays the same, while we have new label sets $S^{\prime}(P)=\{1,2,3,5,6,9\}$ and $S^{\prime}(Q)=\{4,7,8,10\}$, with $3$ and $4$ swapped. On the other hand, if we act on $\ell$ by $t_{5}$, the resulting $S(P),S(Q)$, and $\ell_{Q}$ stay the same while $\ell_{P}^{\prime}=t_{4}(\ell_{P})$.

The following lemma describes the action of $q_{i-1}$ on the label sets $S(P)$ and $S(Q)$.

Lemma 3.15**.**

Let $P,Q$ be finite posets and fix a linear extension $\ell\in{\mathcal{L}}(P+Q)$. Let $1\leq i\leq|P+Q|$, then

1. i)

For all $j\leq i$, $j\in S(P)$ if and only if $i-j+1\in(q_{i-1}S)(P)$; 2. ii)

For all $j>i$, $j\in S(P)$ if and only if $j\in(q_{i-1}S)(P)$.

Proof.

The converse of each statement follows directly from the fact that $q_{i-1}$ is an involution. Note that (ii) is immediate, as $q_{i-1}$ only affects the labels in $[i]$, so $j>i$ remains a label in $(q_{i-1}S)(P)$. It remains to show (i).

Consider the action of $q_{i-1}=\partial_{0}\partial_{1}\partial_{2}\dots\partial_{i-1}$ on a fixed linear extension $\ell\in{\mathcal{L}}(P+Q)$, where $\partial_{j}=t_{j}t_{j-1}\dots t_{1}$ is the promotion operator. Observe that by Definition 2.6, if $1\in S(P)$, then we have $j+1\in(\partial_{j}S)(P)$. On the other hand, if $k\in S(P)$ and $1<k\leq j$, then $k-1\in(\partial_{j}S)(P)$. Let $j\in S(P)$ where $j\leq i$. After applying the first $j-1$ promotion operators in $q_{i-1}$, by this observation, we have $1\in(\partial_{i-j+1}\dots\partial_{i-2}\partial_{i-1}S)(P)$. Thus, by applying the operator $\partial_{i-j}$, we have $i-j+1\in(\partial_{i-j}\dots\partial_{i-2}\partial_{i-1}S)(P)$. Since the label $i-j+1$ is fixed by all subsequent promotions $\partial_{k}$ where $k<i-j$, it follows that $i-j+1\in(q_{i-1}S)(P)$ as desired. ∎

We also need the following lemma describing the action of $q_{jk}$ on $\ell_{P}$ and $\ell_{Q}$.

Lemma 3.16**.**

Let $\ell\in{\mathcal{L}}(P+Q)$, and let $T(\ell)=(\ell_{P},\ell_{Q},S(P),S(Q))$. Then

[TABLE]

where $m=|S(P)\cap[k]|$ and $n=|S(P)\cap[j,k]|-1$.

Proof.

First, we examine the induced action of $q_{jk}$ on $\ell_{P}$. Consider the action of $q_{k-1}=\partial_{0}\partial_{1}\dots\partial_{k-1}$ on $\ell$. The description of promotion in Definition 2.6 implies that each operator $\partial_{i}$ in $q_{k-1}$ ($0\leq i\leq k-1$) only acts on $(\ell_{i})_{P}$ if $\ell_{i}^{-1}(1)$ lies in the component $P$ and on $(\ell_{i})_{Q}$ otherwise, where $\ell_{i}=\partial_{i+1}\dots\partial_{k-1}(\ell)$. The number of promotion operators in $q_{k-1}$ that act on the labels of $P$ therefore is equal to the number of linear extensions $\ell_{i}$ for which $(\ell_{i}^{-1})(P)$ lies in the component $P$, which by the observation in the previous proof is precisely $m=|S(P)\cap[k]|$. Observe that by Definition 2.6, the action of promotion on $\ell_{P}$ does not depend on the labels of $\ell$ on $P$, but only their relative order encoded by $\ell_{P}$. Hence, we deduce that the action on $\ell$ by $q_{k-1}=\partial_{0}\partial_{1}\dots\partial_{k-1}$ is mapped under $T$ to the action on $\ell_{P}$ by $q_{m-1}$.

Now we consider the action of $q_{k-j}$ on $q_{k-1}\ell$. By the same argument above, the induced action of $q_{k-j}$ on $q_{m}(\ell_{P})$ is equivalent to the action of $q_{n^{\prime}-1}$ on $q_{m-1}(\ell_{P})$, where $n^{\prime}=|(q_{k-1}S)(P)\cap[k-j+1]|$. If $c\in S(P)$ and $c>k$, by Lemma 3.15, we have $c\in(q_{k-1}S)(P)$, which does not contribute to $n^{\prime}$. Otherwise, if $c\leq k$, by Lemma 3.15, we have $k-c+1\in(q_{k-1}S)(P)$. Thus, $n^{\prime}=|(q_{k-1}S)(P)\cap[k-j+1]|=|S(P)\cap[j,k]|$.

Finally, we act again by $q_{k-1}$ on $q_{k-j}q_{k-1}\ell$. The induced action of $q_{k-1}$ on $q_{n^{\prime}-1}q_{m-1}\ell_{P}$ is equivalent to the action of $q_{m^{\prime}-1}$ on $q_{n^{\prime}-1}q_{m-1}\ell_{P}$, where $m^{\prime}=|(q_{k-j}q_{k-1}S)(P)\cap[k]|$. However, by Lemma 3.15, we have $|(q_{k-j}q_{k-1}S)(P)\cap[k]|=|(q_{k-1}S)(P)\cap[k]|=|S(P)\cap[k]|$. Thus, the action of $q_{k-1}$ here is equivalent to the action of $q_{m-1}$ on $q_{n^{\prime}-1}q_{m-1}\ell_{P}$, and as the result, the action of $q_{jk}$ on $\ell$ induces the action of $q_{m-1}q_{n^{\prime}-1}q_{m-1}=q_{m-n,m}$ on $\ell_{P}$, where $n=n^{\prime}-1=|S(P)\cap[j,k]|-1$.

By symmetry, the induced action on $\ell_{Q}$ is given by $q_{m^{\prime}-n^{\prime},m^{\prime}}$ where $m^{\prime}=|S(Q)\cap[k]|$ and $n^{\prime}=|S(Q)\cap[j,k]|-1$. Since $|S(P)\cap[k]|+|S(Q)\cap[k]|=k$ and $|S(P)\cap[j,k]|+|S(Q)\cap[j,k]|=k-j+1$, we deduce that $m^{\prime}=k-m$ and $n^{\prime}=k-j-n-1$. Hence

[TABLE]

where $m=|S(P)\cap[k]|$ and $n=|S(P)\cap[j,k]|-1$, as desired. ∎

Now we are ready to prove our main proposition.

Theorem 3.17**.**

If $P$ and $Q$ are LE-cactus, then $P+Q$ is LE-cactus.

Proof.

Let $\ell\in{\mathcal{L}}(P+Q)$ be any linear extension of $P+Q$, and let $T(\ell)=(\ell_{P},\ell_{Q},S(P),S(Q))$. Since the map $T$ is injective, it suffices to show that for all $2\leq i+1<j<k\leq|P+Q|$, $t_{i}$ and $q_{jk}$ commute with respect to each of $\ell_{P},\ell_{Q},S(P)$, and $S(Q)$. We consider two general cases.

Case 1: Assume without loss of generality that $i,i+1\in S(P)$. First we show that $t_{i}$ and $q_{jk}$ commute with respect to $S(P)$ and $S(Q)$. Observe that if $i$ (or $i+1$) is in $S(P)$, by applying Lemma 3.15 three times, we have $i$ (or $i+1$) is also in $(q_{k-1}q_{k-j}q_{k-1}S)(P)=(q_{jk}S)(P)$. It follows that $t_{i}$ acts as the identity on both $S(P)$ and $(q_{jk}S)(P)$, so clearly $t_{i}$ and $q_{jk}$ commute.

It remains to show that $t_{i}$ and $q_{jk}$ commute with respect to $\ell_{P}$ and $\ell_{Q}$. Since $i,i+1\in(q_{jk}S)(P)$, $t_{i}$ does not affect $\ell_{Q}$ and $q_{jk}(\ell_{Q})$. It then suffices to show that $t_{i}$ and $q_{jk}$ commute with respect to $\ell_{P}$. By Lemma 3.16, the induced actions of $t_{i}q_{jk}$ and $q_{jk}t_{i}$ on $\ell_{P}$ are given by to $t_{i^{\prime}}q_{m-n,m}$ and $q_{m-n,m}t_{i^{\prime}}$, respectively, where $m=|S(P)\cap[k]|$ and $n=|S(P)\cap[j,k]|-1$ and $i^{\prime}=|S(P)\cap[i]|$. Further note that

[TABLE]

since $i+1\in S(P)$, thus $i^{\prime}+1<m-n$. Hence, we have $t_{i^{\prime}}q_{m-n,m}(\ell_{P})=q_{m-n,m}t_{i^{\prime}}(\ell_{P})$, following directly from the assumption that $P$ is LE-cactus.

Case 2: Assume without loss of generality that $i\in S(P)$ and $i+1\in S(Q)$, which configuration we simply denote as $(i,i+1)$. We consider where the labels $i,i+1$ are taken under $t_{i}$ and $q_{jk}$. When acting on $\ell$ by $q_{jk}t_{i}$, by the above observation we have $(i,i+1)\xrightarrow{t_{i}}(i+1,i)\xrightarrow{q_{jk}}(i+1,i)$. On the other hand, acting on $\ell$ by $t_{i}q_{jk}$ gives $(i,i+1)\xrightarrow{q_{jk}}(i,i+1)\xrightarrow{t_{i}}(i+1,i)$, which agrees with the action of $q_{jk}t_{i}$. A consequence is that the set of other labels are preserved by both $t_{i}q_{jk}$ and $q_{jk}t_{i}$, and since $t_{i}$ fixes all other labels, we deduce that $t_{i}$ and $q_{jk}$ commute with respect to $S(P)$ and $S(Q)$.

By a prior observation, we have $i\in(q_{jk}S)(P)$ and $i+1\in(q_{jk}S)(Q)$. Thus, $t_{i}$ acts as the identity on each of $\ell_{P},\ell_{Q},q_{jk}(\ell_{P})$, and $q_{jk}(\ell_{Q})$, as swapping elements $i$ and $i+1$ does not change the relative ordering within the label sets. Then clearly $t_{i}$ and $q_{jk}$ commute with respect to $\ell_{P}$ and $\ell_{Q}$.

Hence, we conclude that $t_{i}q_{jk}(\ell)=q_{jk}t_{i}(\ell)$ for all $\ell\in{\mathcal{L}}(P+Q)$, so $P+Q$ is LE-cactus. ∎

The converse of this proposition follows directly from Proposition 3.12. This property then allows us to narrow our focus to studying connected LE-cactus posets.

3.2.2. Ordinal sums of LE-cactus posets

In contrast with disjoint unions, ordinal sums of LE-cactus posets exhibit a more complicated behavior. In particular, the ordinal sum of two LE-cactus posets is not guaranteed to be LE-cactus.

Let $P$ be an $n$-element poset. We next study the construction $P\mapsto A_{m}\oplus P$, where $A_{m}$ is an antichain of $m$ elements, and ask when it preserves the LE-cactus property.

Proposition 3.18**.**

If $P$ is LE-cactus, then $A_{1}\oplus P$ is LE-cactus.

Proof.

Observe that there is a one-to-one correspondence between the sets of linear extensions of $P$ and $A_{1}\oplus P$: a linear extension of $A_{1}\oplus P$ is simply a linear extension of $P$ (with labels shifted up by $1$) with the unique minimal element of $A_{1}\oplus P$ (namely the singleton in $A_{1}$) labelled $1$. Thus $t_{1}$ is the identity. It follows that for any $3\leq i+1<j<k-1\leq n+1$, the relation $(t_{i}q_{jk})^{2}=1$ on ${\mathcal{L}}(A_{1}\oplus P)$ reduces to a cactus relation on linear extensions of the convex induced subposet $P$ of $A_{1}\oplus P$, which holds since $P$ is LE-cactus. The only other cases are when $i=1$ or $k=j+1$. When $i=1$, $(t_{1}q_{jk})^{2}=q_{jk}^{2}=1$ for all $2<j<k\leq n+1$, since $q_{jk}$ is an involution. When $k=j+1$, observe that

[TABLE]

since $q_{k-1}$ is an involution. Thus $q_{jk}=1$, so that $(t_{i}q_{jk})^{2}=t_{i}^{2}=1$ as desired. ∎

Proposition 3.19**.**

If $A_{1}\oplus P$ is LE-cactus, then $A_{2}\oplus P$ is LE-cactus.

Hence if $P$ is LE-cactus, then so are both $A_{1}\oplus P$ and $A_{2}\oplus P$.

Proof.

Suppose $A_{2}=\{p_{1}\}+\{p_{2}\}$, and let $P_{i}$ be the convex induced subposet of $A_{2}\oplus P$ of the form $\{p_{i}\}\oplus P$. Each poset $P_{i}$ is LE-cactus by our assumption. Similar to the previous argument, observe that a linear extension of $A_{2}\oplus P$ is a linear extension of $P$ (with labels shifted up by $2$) with the labels of minimal elements $p_{1}$ and $p_{2}$ drawn from $\{1,2\}$. Thus $t_{2}$ is the identity. For $j\geq 1$, we have

[TABLE]

where

[TABLE]

Since $q_{1}=t_{1}$, by definition $q^{\prime}_{1}=1$. Notice that $t_{1}$ and $q^{\prime}_{j}$ always commute, so for $2<j<k\leq n+2$, we have

[TABLE]

Since each $q^{\prime}_{j}$ commutes with $t_{1}$, so does $q^{\prime}_{jk}$.

We show that the cactus relation $(t_{i}q_{jk})^{2}=1$ holds by considering the following cases:

Case 1: $i=2$. We have $(t_{2}q_{jk})^{2}=q_{jk}^{2}=1$, since $q_{jk}$ is an involution.

Case 2: $i=1$. We have $t_{1}q_{jk}=t_{1}t_{1}^{k-j}q^{\prime}_{jk}=t_{1}^{k-j}q^{\prime}_{jk}t_{1}=q_{jk}t_{1}$, so $(t_{1}q_{jk})^{2}=1$.

Case 3: $i\geq 3$ and $k-j=1$. Recall that $q^{\prime}_{k-j}=q^{\prime}_{1}=1$, so $q_{jk}=t_{1}(q^{\prime}_{k-1})^{2}=t_{1}$, which obviously commutes with $t_{i}$ if $i\geq 3$.

Case 4: $i\geq 3$ and $k-j\geq 2$. We have $(t_{i}q_{jk})^{2}=(t_{i}t_{1}^{k-j}q^{\prime}_{jk})^{2}=(t_{1}^{k-j})^{2}(t_{i}q^{\prime}_{jk})^{2}=(t_{i}q^{\prime}_{jk})^{2}$, so it suffices to show that $(t_{i}q^{\prime}_{jk})^{2}=1$ for all $i\geq 3$ and $k-j\geq 2$. This equation reduces to a cactus relation on linear extensions of either $P_{1}$ or $P_{2}$, which holds since both subposets are LE-cactus by assumption.

The second part follows immediately from Proposition 3.18 and the first part. ∎

Seeing Propositions 3.18, 3.19, one might think that for any LE-cactus poset $P$, the ordinal sum $A_{m}\oplus P$ remains LE-cactus for all $m\geq 1$. Unfortunately, this is not the case; in fact, for $m\geq 3$, taking the ordinal sum with the antichain $A_{m}$ destroys any LE-cactus property $P$ may have.

Proposition 3.20**.**

For any non-empty finite poset $P$, the poset $A_{m}\oplus P$ is not LE-cactus if $m\geq 3$.

Proof.

Observe that $A_{m}\oplus P$ always contains a copy of $A_{m}\oplus A_{1}$ as an order ideal. Thus, by Proposition 3.12, it suffices to show that $A_{m}\oplus A_{1}$ is not LE-cactus for all $m\geq 3$. When $m=3$, one can check that the cactus relation $(t_{1}q_{34})^{2}=1$ does not hold. When $m\geq 4$, we claim that the cactus relation $(t_{1}q_{m-1,m+1})^{2}=1$ does not hold. By studying the evacuation $q_{j}$ on linear extensions of the poset $A_{m}\oplus A_{1}$ (see, e.g., [Sta09]), we observe that $q_{m-1,m+1}$ swaps the labels $1$ and $m-1$, while $t_{1}$ swaps the labels $1$ and $2$. These descriptions show that they do not commute. ∎

While taking the ordinal sum with sufficiently large antichains creates a source of non-LE-cactus posets, the same operation with sufficiently long chains appears to have the exact opposite effect. Let $C_{m}$ denote a chain of $m$ elements.

Proposition 3.21**.**

For any $n$-element poset $P$, $C_{m}\oplus P$ is LE-cactus for all $m\geq n-3$.

Proof.

By Proposition 3.18, it suffices to show that $C_{n-3}\oplus P$ is LE-cactus. The key observation is that $t_{i}$ is the identity for all $i\leq n-3$, since the element of $P$ labelled by $i$ lies inside the lower chain and thus is comparable to that labelled by $i+1$. Consider the operators $t_{i}$ and $q_{jk}=q_{k-1}q_{k-j}q_{k-1}$ for $2\leq i+1<j<k\leq 2n-3$. If $i\leq n-3$, $t_{i}=1$ obviously commutes with $q_{jk}$. If $i\geq n-2$, then $j\geq n$, so $k-j\leq(2n-3)-n=n-3$. It follows that $q_{k-j}=t_{1}(t_{2}t_{1})\ldots(t_{k-j}t_{k-j-1}\ldots t_{1})=1$, therefore $q_{jk}=q_{k-1}^{2}=1$ which obviously commutes with $t_{i}$. ∎

3.3. Minuscule, d-complete, and jeu-de-taquin posets

Starting with the singleton poset $A_{1}$, by iterating the constructs of disjoint union and ordinal sum $P\mapsto A_{1}\oplus P$, we produce the family of all rooted forest posets–disjoint unions of trees with a unique minimal element, which are LE-cactus by Theorem 3.17 and Proposition 3.18. Knuth observed a famous hook-length formula for counting the linear extensions of such forest posets [Knu73]. The original motivation for this finding was the family of Ferrers posets for which the Frame–Robinson–Thrall hook-length formula counts their linear extensions [FRT54], another family of posets that we know are LE-cactus. Motivated by these two families with hook formulas, Proctor introduced the family of d-complete posets having such a hook-length formula for their linear extensions; we refer the readers to [PS19] and [KY19] for the precise definitions and hook-length formulas. This raises the question of whether all d-complete posets are LE-cactus.

Preliminary investigations point to an affirmative answer. Besides the above families, it is not too hard to show that all shifted Ferrers posets, another motivating subfamily of d-complete posets, are LE-cactus. The idea is to embed standard shifted tableaux, which are in bijection with linear extensions of shifted Ferrers posets, by “doubling” them into a shift-symmetric column-strict tableaux [Sag87]; for example,

[TABLE]

Then, the action of the usual BK moves $t_{i}$ on the standard shifted tableaux commute with the action of $t_{i}$ on the shift-symmetric tableaux, where we know that BK moves satisfy the cactus relations.

Another closely related family of posets is the minuscule posets, which first appeared in Lie theory. Proctor’s d-complete posets contain all order ideals within minuscule posets as a motivating special case. Minuscule posets are classified into three infinite families and two exceptional cases (see, e.g., [Ste90, RS13]). The infinite families of minuscule posets are the rectangular Ferrers posets (whose order ideals contain all Ferrers posets), the triangular shifted Ferrers posets (whose order ideals contain all shifted Ferrers posets), and ordinal sums of antichains $A_{1}^{\oplus n}\oplus A_{2}\oplus A_{1}^{\oplus n}$; all of which are LE-cactus based on our results so far. By checking the LE-cactus property of the remaining exceptional minuscule posets using SageMath, we have verified the following:

Theorem 3.22**.**

Minuscule posets are LE-cactus.

Based on the existing evidence, we make the following conjecture.

Conjecture 3.23**.**

d-complete posets are LE-cactus.

This has been verified by SageMath for all d-complete posets of size at most $9$.

Remark 3.24*.*

In [Pro09], Proctor also studied a large family of posets called jeu-de-taquin posets, which he showed includes all d-complete posets. Informally, jeu-de-taquin posets are posets for which Schützenberger’s jeu-de-taquin on linear extensions of posets, as discussed in [Sta09], has an extra confluence property.

For such posets, we could show that $q_{jk}$ only permutes the labels $j,j+1,\dots,k$ when acting on a linear extension. Thus, it is more probable for $t_{i}$ and $q_{jk}$ to commute because they affect disjoint subsets of the labels. Hence, one might hope that jeu-de-taquin posets are also LE-cactus. However, this is not true; Figure 2 shows a counterexample to the LE-cactus property of jeu-de-taquin posets, where the cactus relation $(t_{3}q_{59})^{2}=1$ does not hold for this linear extension whose underlying poset is jeu-de-taquin.

4. LE-symmetric posets

In this section, we will study the group $\mathcal{BK}_{P}$ as a subgroup of the symmetric group on the set ${\mathcal{L}}(P)$. In particular, we focus on understanding posets $P$ for which $\mathcal{BK}_{P}={\mathfrak{S}}_{{\mathcal{L}}(P)}$.

Definition 4.1**.**

A finite poset $P$ is called LE-symmetric if $\mathcal{BK}_{P}={\mathfrak{S}}_{{\mathcal{L}}(P)}$.

4.1. Duals and ordinal sums

As in Section 3.2, we explore properties of LE-symmetric posets, particularly their behavior under poset operations such as taking duals and ordinal sums. Let $P$ and $Q$ be finite posets, and let $P^{*}$ denote the dual or opposite poset to $P$.

Proposition 4.2**.**

One has a group isomorphism $\mathcal{BK}_{P}\cong\mathcal{BK}_{P^{*}}$.

Hence $P$ is LE-symmetric if and only if $P^{*}$ is LE-symmetric.

Proof.

One has a bijection ${\mathcal{L}}(P)\to{\mathcal{L}}(P^{*})$ sending $\ell\mapsto\ell^{\prime}$ where $\ell^{\prime}(v)=|P|+1-\ell(v)$ for all $v\in P$. This induces an isomorphism of symmetric groups ${\mathfrak{S}}_{{\mathcal{L}}(P)}\rightarrow{\mathfrak{S}}_{{\mathcal{L}}(P^{*})}$, which then restricts to an isomorphism $\mathcal{BK}_{P}\to\mathcal{BK}_{P^{*}}$ sending $t_{i}\mapsto t_{|P|-i}$. ∎

Likewise, the BK group of the ordinal sum of two posets has a very nice description in terms of the BK groups of its components.

Proposition 4.3**.**

$\mathcal{BK}_{P\oplus Q}\cong\mathcal{BK}_{P}\times\mathcal{BK}_{Q}$.

Proof.

Let

[TABLE]

where $\ell_{1}$ is $\ell$ restricted to $P$ and $\ell_{2}$ is $\ell-|P|$ restricted to $Q$. We have that $\lambda$ is a bijection. Now let

[TABLE]

where

[TABLE]

We have that $\psi$ is an isomorphism of permutation groups as desired. ∎

This property leads to some useful tools for constructing and identifying LE-symmetric posets that arise from ordinal sums.

Corollary 4.4**.**

A poset $P$ is LE-symmetric if and only if $C_{a}\oplus P\oplus C_{b}$ is LE-symmetric for any chains $C_{a}$ and $C_{b}$.

Proof.

This corollary follows directly from the previous proposition and the fact that for any chain $C_{a}$, the set ${\mathcal{L}}(C_{a})$ is a singleton, so its BK group is trivial. ∎

Corollary 4.5**.**

If $P$ and $Q$ are posets such that $P\oplus Q$ is LE-symmetric, then at least one of them is a chain. As a consequence, $P$ and $Q$ are LE-symmetric.

Proof.

Since $P\oplus Q$ is LE-symmetric, $\mathcal{BK}_{P\oplus Q}$ is a symmetric group. By Proposition 4.3, it can be written as a direct product of the ordinal summands’ BK groups, i.e., $\mathcal{BK}_{P\oplus Q}~{}\cong~{}\mathcal{BK}_{P}\times\mathcal{BK}_{Q}$. Then at least one of the factors $\mathcal{BK}_{P}$ and $\mathcal{BK}_{Q}$ is the trivial group. This happens when at least one of the posets $P$ and $Q$ has exactly one linear extension and hence is a chain. The second part follows immediately from the previous corollary. ∎

Corollaries 4.4 and  4.5 are especially useful in identifying LE-symmetric posets, as they allow us to turn our attention to posets with neither a unique maximal nor minimal element.

4.2. LE-primitive posets and disjoint unions

The goal of this subsection is to prove a classification theorem for disconnected LE-symmetric posets. En route to this result, we will first study a generalization of LE-symmetric posets. Recall that the action of a group $G$ on a set $X$ is primitive if there is no partition of $X$ that is preserved by $G$ other than the trivial partitions (i.e., the finest and coarsest partitions). In this subsection, we turn our attention to posets whose BK groups are primitive.

Definition 4.6**.**

A poset $P$ is LE-primitive if $\mathcal{BK}_{P}$ is a primitive permutation group.

This concept generalizes the LE-symmetric posets, since ${\mathfrak{S}}_{{\mathcal{L}}(P)}$ is a primitive permutation group. We will give a classification of disconnected LE-primitive posets.

Lemma 4.7**.**

If an $n$-element poset $P$ is a disjoint union of two or more non-empty chains, then $\mathcal{BK}_{P}\cong{\mathfrak{S}}_{n}$.

Proof.

By Proposition 3.8, $\mathcal{BK}_{P}$ is isomorphic to a quotient of ${\mathfrak{S}}_{n}$. Thus, it suffices to show that the group is not trivial, is not $C_{2}$ for $n\geq 3$, and is not ${\mathfrak{S}}_{3}$ for $n=4$. Each of these is easily checked. ∎

Lemma 4.8**.**

Let $P=C_{n_{1}}+C_{n_{2}}+\dots+C_{n_{r}}$. Then, ${\operatorname{Stab}}(\ell)\cong{\mathfrak{S}}_{n_{1}}\times{\mathfrak{S}}_{n_{2}}\times\dots\times{\mathfrak{S}}_{n_{r}}$ for any $\ell\in{\mathcal{L}}(P)$.

Proof.

We have ${\operatorname{Stab}}(\ell_{1})\cong{\operatorname{Stab}}(\ell_{2})$ for any $\ell_{1},\ell_{2}\in{\mathcal{L}}(P)$, so it suffices to consider only one linear extension. Let $\ell$ be the linear extension that assigns consecutive labels to each chain beginning with $C_{n_{1}}$. Then, $t_{i}\in{\operatorname{Stab}}(\ell)$ if and only if $i\neq\sum_{j=1}^{k}n_{j}$ for all $1\leq k\leq r-1$. Recall that by Lemma 4.7, $t_{1},\dots,t_{n-1}$ yield a presentation for the symmetric group ${\mathfrak{S}}_{n}$, where

[TABLE]

Thus, the group generated by $\{t_{i}\mid i\neq\sum_{j=1}^{k}n_{j}\ \text{for all}\ 1\leq k\leq r-1\}$ is isomorphic to ${\mathfrak{S}}_{n_{1}}\times{\mathfrak{S}}_{n_{2}}\times\dots\times{\mathfrak{S}}_{n_{r}}$. It remains to show that ${\mathfrak{S}}_{n_{1}}\times{\mathfrak{S}}_{n_{2}}\times\dots\times{\mathfrak{S}}_{n_{r}}$ exhausts ${\operatorname{Stab}}(\ell)$, which follows from the following calculation:

[TABLE]

∎

Lemma 4.9**.**

If $P=P_{1}+P_{2}$, where $|P_{1}|,|P_{2}|\geq 1$ and $P_{1}$ is not a chain, then $P$ is not LE-primitive.

Proof.

Given $\ell\in{\mathcal{L}}(P)$, assign $\ell$ the set $\ell(P_{1})$. This induces a partition on ${\mathcal{L}}(P)$. The conditions of the lemma imply that the partition is nontrivial. It’s easy to check that each $t_{i}$ preserves the partition. ∎

Theorem 4.10**.**

A disconnected poset $P$ is LE-primitive if and only if $P=C_{n_{1}}+C_{n_{2}}$, where $n_{1}\neq n_{2}$ or $n_{1}=n_{2}=1$.

Proof.

First, if one component of $P$ is not a chain, then $P$ is not LE-primitive by Lemma 4.9. If $P=C_{n_{1}}+C_{n_{2}}+\dots+C_{n_{r}}$, then $\mathcal{BK}_{P}\cong{\mathfrak{S}}_{n}$ by Lemma 4.7 and ${\operatorname{Stab}}(\ell)\cong{\mathfrak{S}}_{n_{1}}\times{\mathfrak{S}}_{n_{2}}\times\dots\times{\mathfrak{S}}_{n_{r}}$ for any $\ell\in{\mathcal{L}}(P)$ by Lemma 4.8. We have that ${\mathfrak{S}}_{n_{1}}\times{\mathfrak{S}}_{n_{2}}\times\dots\times{\mathfrak{S}}_{n_{r}}$ is a maximal proper subgroup of ${\mathfrak{S}}_{n}$ if and only if $r=2$ and either $n_{1}\neq n_{2}$ or $n_{1}=n_{2}=1$. Since a permutation group is primitive if and only if the stabilizer of any element is a maximal subgroup, this completes the proof. ∎

The first application of this theorem is a classification of LE-primitive series-parallel posets, i.e., posets that can be constructed from the singleton $A_{1}$ using the ordinal sum and disjoint union operations.

Corollary 4.11**.**

A non-empty series-parallel poset $P$ is LE-primitive if and only if $P=C_{n_{1}}\oplus(C_{n_{2}}+C_{n_{3}})\oplus C_{n_{4}}$, where $|n_{2}|\neq|n_{3}|$ or $|n_{2}|=|n_{3}|=1$.

Proof.

Every poset of the stated form is series-parallel and LE-primitive by Theorem 4.10. Conversely, let $P$ be an LE-primitive non-empty series-parallel poset. If $P=P_{1}+P_{2}$, where $|P_{1}|,|P_{2}|\geq 1$, then $P=C_{n_{2}}+C_{n_{3}}$ with the stated conditions by Theorem 4.10. If $P=P_{1}\oplus P_{2}$, where $|P_{1}|,|P_{2}|\geq 1$, then $P_{1}=C_{n_{1}}$ or $P_{2}=C_{n_{4}}$ by Proposition 4.3. Thus, $P$ is of the desired form. ∎

We now return to disconnected LE-symmetric posets. The proof of our classification theorem in this case follows a similar logic as that of Theorem 4.10.

Theorem 4.12**.**

A disconnected poset $P$ is LE-symmetric if and only if $P=C_{n_{1}}+A_{1}$.

Proof.

First, if one component of $P$ is not a chain, then $P$ is not LE-symmetric by Lemma 4.9. If $P=C_{n_{1}}+C_{n_{2}}+\dots+C_{n_{r}}$, then $\mathcal{BK}_{P}\cong{\mathfrak{S}}_{n}$ by Lemma 4.7. Thus, $P$ is LE-symmetric if and only if $|{\mathcal{L}}(P)|=n$, which happens if and only if $P$ is of the stated form. ∎

Corollary 4.13**.**

A non-empty series-parallel poset $P$ is LE-symmetric if and only if $P=C_{n_{1}}\oplus(C_{n_{2}}+A_{1})\oplus C_{n_{3}}$.

Proof.

Every poset of the form $C_{n_{1}}\oplus(C_{n_{2}}+A_{1})\oplus C_{n_{3}}$ is series-parallel and LE-symmetric by Theorem 4.12. Conversely, let $P$ be an LE-symmetric non-empty series-parallel poset. If $P=P_{1}+P_{2}$, where $|P_{1}|,|P_{2}|\geq 1$, then $P=C_{n_{2}}+A_{1}$ by Theorem 4.12. If $P=P_{1}\oplus P_{2}$, where $|P_{1}|,|P_{2}|\geq 1$, then $P_{1}=C_{n_{1}}$ or $P_{2}=C_{n_{4}}$ by Proposition 4.3. Thus, $P$ is of the desired form. ∎

4.3. LE-symmetric families

We next study LE-symmetric non-series-parallel posets. Theorem 4.12 allows us to restrict our study to those which are connected. The following class of posets presents a suggestive starting point:

Definition 4.14**.**

For integers $a,b,c\geq 1$, the poset $N_{a,b,c}$ is defined to be a poset of $a+b+c$ elements with the relations

[TABLE]

In particular, every non-empty non-series-parallel poset contains a poset $N_{1,b,1}$ as a convex induced subposet.

Proposition 4.15**.**

The poset $N_{1,b,1}$ is LE-symmetric for all $b\geq 1$.

Proof.

Let $n=b+3$, so $N_{1,b,1}=v_{1}<v_{2}>v_{3}>\ldots>v_{n-2}>v_{n-1}<v_{n}$.

For $n\in\{4,5\}$, the theorem is verified by a calculation. So suppose $n\geq 6$.

For $i,j\in[n]$, there is at most one linear extension $\ell$ with $\ell(v_{1})=i$ and $\ell(v_{n})=j$. If it exists, we denote it $\ell_{i,j}$.

First, we claim $(t_{2}t_{3})^{3}$ and $(t_{n-3}t_{n-2})^{3}$ are the transpositions $(\ell_{3,1}\;\ell_{4,1})$ and $(\ell_{n,n-3}\;\ell_{n,n-2})$ respectively. This follows from the characterization of when the braid relations hold.

Now, it suffices to prove that $N_{1,b,1}$ is LE-2-transitive (i.e., $\mathcal{BK}_{N_{1,b,1}}$ is 2-transitive), which is equivalent to its point-stabilizers’ being transitive. Consider ${\operatorname{Stab}}(\ell_{n,1})$. Note that $t_{2},t_{3},\dots,t_{n-2}\in{\operatorname{Stab}}(\ell_{n,1})$. Hence, ${\operatorname{Stab}}(\ell_{n,1})$ is transitive on each of $\{\ell_{i,j}\mid i\leq n-1,\ j\geq 2\}$, $\{\ell_{i,j}\mid i\leq n-1,\ j=1\}$, and $\{\ell_{i,j}\mid i=n,\ j\geq 2\}$. It remains to observe that

[TABLE]

so ${\operatorname{Stab}}(\ell_{n,1})$ is transitive. ∎

Proposition 4.16**.**

The poset $N_{1,1,c}$ is LE-symmetric for all $c\geq 1$, and the poset $N_{a,1,1}$ is LE-symmetric for all $a\geq 1$.

Proof.

By duality, it suffices to prove the result for $N_{1,1,c}$. Let $n=c+3$, so $N_{1,1,c}=v_{1}<v_{2}>v_{3}<v_{4}<\dots<v_{n}$.

For $i\in[n-1]$ and $j\in[\max\{3,i+1\},n]$, there is exactly one linear extension $\ell$ with $\ell(v_{1})=i$ and $\ell(v_{n})=j$, which we denote $\ell_{i,j}$.

First, we claim $(t_{2}t_{3})^{3}=(\ell_{1,3}\;\ell_{1,4})$. This follows from the characterization of when the braid relations hold.

Now, it suffices to prove that $N_{1,1,c}$ is LE-2-transitive, which is equivalent to its point-stabilizers’ being transitive. Consider ${\operatorname{Stab}}(\ell_{n-1,n})$. Note that $t_{1}$, $t_{2},\dots,t_{n-3}$, $t_{n-1}\in{\operatorname{Stab}}(\ell_{n-1,n})$. Hence, ${\operatorname{Stab}}(\ell_{n-1,n})$ is transitive on $\{\ell_{i,j}\mid j\leq n-2\}$ and $\{\ell_{i,j}\mid j\geq n-1,\ i\leq n-1\}$. It remains to observe that

[TABLE]

so ${\operatorname{Stab}}(\ell_{n-1,n})$ is transitive. ∎

By using the same proof strategy, we are able to prove LE-symmetry for a variant of these posets.

Proposition 4.17**.**

For $a,b\geq 2$, let $M_{a,b}$ be a poset on $a+b$ elements with the relations $v_{1}<v_{2}<\dots<v_{a}$, $v_{a+1}<v_{a+2}<\dots<v_{a+b}$, and $v_{a-1}<v_{a+2}$ (see Figure 4). Then $M_{a,b}$ is LE-symmetric.

Proof.

First, let $\ell^{*}$ be the linear extension such that $\ell^{*}(v_{i})=i$. Notice that

[TABLE]

Further note that $t_{a}\notin{\operatorname{Stab}}(\ell)$ if and only if $\ell\in\{\ell^{*},t_{a}(\ell^{*})\}$, i.e., $t_{a}$ acts as the transposition $(\ell^{*}\;t_{a}(\ell^{*}))$ on ${\mathcal{L}}(M_{a,b})$.

Thus, it suffices to show that ${\operatorname{Stab}}(\ell^{*})$ is transitive on ${\mathcal{L}}(M_{a,b})\setminus\{\ell^{*}\}$. This follows from the observations above and Proposition 2.8, which states that $\mathcal{BK}_{P}$ is a transitive group. ∎

One way to generalize the poset $M_{a,b}$ is to replace the covering relation $v_{a-1}<v_{a+2}$ with another relation $x<y$ where $x$ and $y$ are drawn from two disjoint chains. These posets, however, are not always LE-symmetric; for example, we found a counterexample in the poset defined by the relations $v_{1}<v_{2}<v_{3}<v_{4}$, $v_{5}<v_{6}<v_{7}<v_{8}$, and $v_{2}<v_{7}$.

4.3.1. Conjectural LE-symmetric families and computational results

A natural conjectural LE-symmetric family of posets follows from our investigation above.

Conjecture 4.18**.**

The poset $N_{a,b,c}$ is LE-symmetric for all $a,b,c\geq 1$.

Besides the subfamilies proved in Propositions 4.15 and 4.16, our preliminary computation by SageMath indicates that the poset $N_{a,b,c}$ is LE-symmetric for all $1\leq a,b,c\leq 4$. As a remark, the proofs of Propositions 4.15,  4.16, and  4.17 rely on an observational analysis of the linear extension graphs of these posets. As the parameters $(a,b,c)$ grow, the linear extension graph of the poset $N_{a,b,c}$ becomes significantly more complex; for example, its dimension increases as parameters $a$ and $c$ grow, and it is no longer apparent that we can generate a transposition on ${\mathcal{L}}(N_{a,b,c})$ from BK moves. A different proof strategy may be required for the general case.

Another family of posets that is closely related to $N_{a,b,c}$ is also conjectured to be LE-symmetric.

Conjecture 4.19**.**

The zigzag-poset $Z_{n}$, defined to be a poset of $n$ elements with the relations

[TABLE]

is LE-symmetric when $n$ is even.

The zigzag-poset $Z_{n}$ is computationally checked by SageMath to be LE-symmetric for all even $n\leq 10$. When $n$ is odd, we found small counterexamples in $Z_{5},Z_{7}$, and $Z_{9}$. A potential common generalization of Conjectures 4.18 and 4.19 concerns the LE-symmetric property of zig-zag posets of the form $Z_{a_{1},\dots,a_{n}}$ for odd $n$, where $a_{i}$ is the length of the $i^{\mathrm{th}}$ segment; however, we found a counterexample in $Z_{1,2,2,2,2}$ which is not LE-symmetric. Finally, we remark that the linear extension graphs of zig-zag posets are very complicated and high-dimensional, for which our observational analysis used in the proof of Proposition 4.15 is not suitable.

The final conjectural LE-symmetric family of posets involves the Ferrers posets which, in contrast with our finding of their LE-cactus property, are not always LE-symmetric. Note that in a slightly different language, Vershik and Tsilevich showed that $F_{\lambda}$ is LE-primitive when $\lambda=(n,1,\dots,1)$ or $\lambda=(n,2)$, and gave computational evidence for small Ferrers posets that their BK groups should often be primitive or at least Coxeter [VT19]. Observe that for any partition $\lambda$ and its conjugate $\lambda^{t}$, $F_{\lambda}\cong F_{\lambda^{t}}$ are isomorphic posets.

Conjecture 4.20**.**

The families of Ferrers posets $F_{\lambda}$, and therefore $F_{\lambda^{t}}$, are LE-symmetric for the following partition $\lambda$:

1. (1)

$\lambda=(n,n-2)$* for all $n$;* 2. (2)

$\lambda=(n,3)$* for $n\not\equiv 2\pmod{4}$; and* 3. (3)

$\lambda=(n,2,2)$* for $n\not\equiv 0\pmod{4}$.*

These conjectural classes have been computationally checked by SageMath to be LE-symmetric for (1) $n\leq 10$, (2) $n\leq 18$, and (3) $n\leq 16$. The data set for this analysis was generously provided by J. Kamnitzer.

5. Size of BK groups

In this final section, we explore two different notions of size of the group $\mathcal{BK}_{P}$.

5.1. Duals, ordinal sums, and disjoint unions

We explore the cardinality of BK groups of duals $P^{*}$, ordinal sums $P\oplus Q$, and disjoint unions $P+Q$ relative to those of $P$ and $Q$. The cardinality of duals and ordinal sums follows immediately from Propositions 4.2 and 4.3:

Proposition 5.1**.**

Let $P$ and $Q$ be finite posets.

1. (1)

$|\mathcal{BK}_{P^{*}}|=|\mathcal{BK}_{P}|$; 2. (2)

$|\mathcal{BK}_{P\oplus Q}|=|\mathcal{BK}_{P}||\mathcal{BK}_{Q}|$.

As for disjoint unions, the following proposition gives an upper bound for the size of the permutation group $\mathcal{BK}_{P+Q}$.

Proposition 5.2**.**

Let $|P|=m$ and $|Q|=n$. Then

[TABLE]

Proof.

We bound $|\mathcal{BK}_{P+Q}|$ by directly bounding the number of permutations on ${\mathcal{L}}(P+Q)$ we can obtain.

For any partition $K=(A,B)$, define the projection map

[TABLE]

(see Definition 3.13 for the definition of $T$). We define the partition set to be $L(K)=(\pi_{S}\circ T)^{-1}(K)$, i.e., the set of linear extensions whose labels of $P$ are in $A$ and whose labels of $Q$ are in $B$.

Since the action of $t_{i}$ preserves each partition set $L(K)$, we have that the total number of permutations on ${\mathcal{L}}(P+Q)$ is bounded by the product of the number of permutations on the $\binom{m+n}{n}$ partition sets (treating each partition set as an element) and the number of permutations that preserve all partition sets.

First, we count the number of ways to permute the $\binom{m+n}{n}$ partition sets. Note that the action of the $t_{i}$ on the partition sets themselves is isomorphic to the action of $t_{i}$ the poset $P=C_{m}+C_{n}$, i.e., the disjoint union of a chain of size $m$ and a chain of size $n$. To see this, any partition set $K=(A,B)$ corresponds to the linear extension $\ell$ on $C_{m}+C_{n}$ such that $\ell(C_{m})=A$ and $\ell(C_{n})=B$. By Lemma 4.7, this group is precisely ${\mathfrak{S}}_{n+m}$, so the number of ways to permute the partition sets is given by $|{\mathfrak{S}}_{m+n}|=(m+n)!$.

Now we count the number of permutations that preserve the partition sets. Let $\mathcal{BK}_{K}$ be the subgroup of permutations of $\mathcal{BK}_{P+Q}$ which preserve $L(K)$. Then for each permutation of the partition sets, the number of permutations that preserve all partition sets is bounded above by $\prod_{K}|\mathcal{BK}_{K}|$.

We claim that $|\mathcal{BK}_{K}|=|\mathcal{BK}_{K^{\prime}}|$, where $K^{\prime}=([m],[m+1,m+n])$. Let $w\in\mathfrak{S}_{m+n}$ be the unique permutation that is order-preserving on $A$ and $B$, with $w(A)=[m]$ and $w(B)=[m+1,m+n]$. Then any nontrivial permutation $\pi$ in $\mathcal{BK}_{K}$ corresponds to a nontrivial permutation $\pi^{\prime}\in\mathcal{BK}_{K^{\prime}}$, where $\pi^{\prime}=w^{-1}\pi w$, and this correspondence is bijective. So, $|\mathcal{BK}_{K}|=|\mathcal{BK}_{K^{\prime}}|$.

Now we compute $\mathcal{BK}_{K^{\prime}}$. The number of ways to permute within $L([m],[m+1,m+n])$ is the number of ways to permute within $L([m],[m+1,m+n])$ without using $t_{m}$. This permutation group, which we called $\mathcal{BK}_{K^{\prime}}$, is then isomorphic to $\mathcal{BK}_{P}\times\mathcal{BK}_{Q}$ with size $|\mathcal{BK}_{P}||\mathcal{BK}_{Q}|$. Since there are $\binom{m+n}{n}$ partition sets, we have

[TABLE]

Thus $|\mathcal{BK}_{P+Q}|\leq(|\mathcal{BK}_{P}||\mathcal{BK}_{Q}|)^{\binom{m+n}{n}}(m+n)!$, as desired. ∎

5.2. Stabilizer size of linear extensions under BK action

In this final subsection, we discuss a different notion of size for the BK group $\mathcal{BK}_{P}$ by investigating the order of the stabilizer of any linear extension of $P$.

Definition 5.3**.**

For a poset $P$, its stabilizer size $\mathfrak{s}_{P}$ is defined to be

[TABLE]

Our naming is suggestive: Since the subgroup $\mathcal{BK}_{P}$ of ${\mathfrak{S}}_{{\mathcal{L}}(P)}$ is transitive, by the orbit-stabilizer theorem, $\mathfrak{s}_{P}$ is the size of the stabilizer of any linear extension of $P$. In particular, $\mathfrak{s}_{P}$ is a positive integer.

Part (1) of Proposition 5.1 and the fact that $|{\mathcal{L}}(P\oplus Q)|=|{\mathcal{L}}(P)|\cdot|{\mathcal{L}}(Q)|$ immediately imply the following.

Proposition 5.4**.**

For any posets $P$ and $Q$, one has $\mathfrak{s}_{P\oplus Q}=\mathfrak{s}_{P}\mathfrak{s}_{Q}.$

We now consider what integers are possible values of $\mathfrak{s}_{P}$. If

[TABLE]

where $C_{n_{i}}$ is a chain of size $n_{i}$, then $\mathfrak{s}_{P}=n_{1}!n_{2}!\dots n_{k}!$ by Lemma 4.8. In particular, any integer that can be expressed as a product of factorials (a Jordan–Pólya number) is a possible value of $\mathfrak{s}_{P}$. But this condition is not necessary. For example, $(A_{3}\oplus A_{1}\oplus A_{1})+A_{1}$ has $\mathfrak{s}_{P}=466560$. Thus, we turn to establishing some necessary conditions on possible values of $\mathfrak{s}_{P}$.

Recall that a poset is indecomposable if it is not an ordinal sum of two or more non-empty posets.

Lemma 5.5**.**

If $P$ is indecomposable and not a disjoint union of chains, then ${\mathfrak{S}}_{4}\leq{\operatorname{Stab}}(\ell)$.

Proof.

Since $P$ is not a disjoint union of chains, $P$ either has a V-shaped or inverted V-shaped induced subposet. By duality, we may assume $P$ has an inverted V-shaped induced subposet; that is, there are $u,v,w\in P$ with the only relations among them being $v<u$ and $w<u$. Since $P$ is indecomposable, there is an immediate child of $u$ incomparable to some $x$, where $x$ is incomparable with $u$. Without loss of generality, assume this immediate child is $w$. If $x$ and $v$ are incomparable, then $u$, $v$, $w$, and $x$ form a convex induced subposet of $P$ with ${\operatorname{Stab}}(\ell)\cong\mathbb{Z}/2\mathbb{Z}\times{\mathfrak{S}}_{4}$. If $x>v$, then $u$, $v$, $w$, and $x$ form a convex induced subposet of $P$ with ${\operatorname{Stab}}(\ell)\cong{\mathfrak{S}}_{4}$. In either case, the proposition follows by Corollary 3.7. ∎

Theorem 5.6**.**

If $\mathfrak{s}_{P}\neq 6,12,36,2^{n}$, then either $(\mathbb{Z}/2\mathbb{Z})^{2}\times{\mathfrak{S}}_{3}\leq{\operatorname{Stab}}(\ell)$ or ${\mathfrak{S}}_{4}\leq{\operatorname{Stab}}(\ell)$. In particular, $\mathfrak{s}_{P}=6,12,36,2^{n}$ or $24\mid\mathfrak{s}_{P}$.

Proof.

We may assume that $P$ is indecomposable. If $P$ is a disjoint union of chains, the theorem follows from Lemma 4.8. Otherwise, the theorem follows from Lemma 5.5. ∎

If $24\nmid\mathfrak{s}_{P}$, then by Lemma 5.5

[TABLE]

where $P_{i}$ is a disjoint union of chains. Then, by Proposition 5.4,

[TABLE]

where each $\mathfrak{s}_{P_{i}}$ can be calculated with Lemma 4.8. This yields classifications of posets with a given $\mathfrak{s}_{P}$ not divisible by 24. As a special case, we have the following proposition.

Proposition 5.7**.**

We have $\mathfrak{s}_{P}=1$ (i.e., the group action is simply transitive) if and only if $P$ is an ordinal sum of antichains.

We now define a poset parameter that gives us useful information about the stabilizer, including a lower bound for $\mathfrak{s}_{P}$.

Definition 5.8**.**

For a poset $P$ and $\ell\in{\mathcal{L}}(P)$, define

[TABLE]

Then, the comparability of $P$ is

[TABLE]

In other words, $c(P)$ is the maximum number of $t_{i}$ such that $t_{i}\in{\operatorname{Stab}}(\ell)$.

Definition 5.9**.**

Let $G$ be a group. We say a set $S\subseteq G$ is independent if $s\notin\langle S\setminus\{s\}\rangle$ for all $s\in S$. Then, let $m(G)$ be the maximum size of an independent set of involutions in $G$.

Proposition 5.10**.**

We have $m(G)\leq\log_{2}(|G|)$.

Proof.

Given an independent set $\{g_{1},\dots,g_{m(G)}\}\subseteq G$, we have

[TABLE]

where each subgroup is at least twice as large as the previous one by Lagrange’s theorem. ∎

Proposition 5.11**.**

If $P$ is indecomposable, then $c(P)\leq m({\operatorname{Stab}}(\ell))$.

Proof.

By Corollary 3.3, $\ell$ has at least $c(P)$ independent involutions in ${\operatorname{Stab}}(\ell)$. ∎

Corollary 5.12**.**

If $P$ is indecomposable, then $\mathfrak{s}_{P}\geq 2^{c(P)}$.

Proof.

This immediate from Propositions 5.10 and 5.11. ∎

Proposition 5.13**.**

We have $h(P)-1\leq c(P)\leq|P|-w(P)$, where $h(P)$ is the height of $P$ and $w(P)$ is the width of $P$. As a consequence, $\mathfrak{s}_{P}\geq 2^{h(P)-1}$.

Proof.

For the first inequality, let $P^{\prime}$ be a subposet of $P$ obtained by removing a maximum chain $C$ from $P$, so $|P^{\prime}|=|P|-h(P)$. We construct $\ell\in{\mathcal{L}}(P)$ as follows. At each step, if label $i$ can be assigned to the least element in $C$ without a label, then do so; otherwise, assign the label $i$ to any other element for which it is possible. If $\ell^{-1}(i),\ell^{-1}(i+1)\in C$, then $\ell^{-1}(i)<\ell^{-1}(i+1)$. Further, if $\ell^{-1}(i)\in P^{\prime}$ and $\ell^{-1}(i+1)\in C$, then $\ell^{-1}(i)<\ell^{-1}(i+1)$. It follows that $h(P)-1\leq c(P)$.

For the second inequality, let $A$ be a maximum antichain in $P$. Let $\ell\in{\mathcal{L}}(P)$. Then, given $x,y\in A$ with $\ell(x)<\ell(y)$, there must be some $i\in[\ell(x),\ell(y)-1]$ such that $\ell^{-1}(i)$ is incomparable to $\ell^{-1}(i+1)$. It follows that $c(P)\leq|P|-w(P)$. ∎