On fully commutative elements of type $\tilde B$ and $\tilde D$

TL;DR

This paper classifies fully commutative elements in affine Coxeter groups of types B and D, constructs towers of these groups and their algebras, and proves faithfulness of the resulting algebraic structures.

Contribution

It introduces a normal form for fully commutative elements in affine types B and D and establishes faithful towers of Coxeter groups and related algebras.

Findings

Normal form classification for fully commutative elements

Construction of faithful towers of Coxeter groups

Faithfulness of affine Temperley-Lieb algebra towers

Abstract

We define a tower of injections of $\tilde{B}$-type (resp. $\tilde{D}$-type) Coxeter groups $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$) for $n\geq 3$. Let $W^c(\tilde B_{n})$ (resp. $W^c(\tilde D_{n})$) be the set of fully commutative elements in $W(\tilde B_{n})$ (resp. $W(\tilde D_{n})$), we classify the elements of this set by giving a normal form for them. We define a $\tilde{B}$-type tower of Hecke algebras and we use the faithfulness at the Coxeter level to show that this last tower is a tower of injections. We use this normal form to define two injections from $W^c(\tilde B_{n-1})$ into $W^c(\tilde B_{n})$. We then define the tower of affine Temperley-Lieb algebras of type $\tilde{B }$ and use the injections above to prove the faithfulness of this tower. We follow the same track for $\tilde{D}$-type objects

Equations

$${\sigma }_{\bar{1}}^2={\sigma }_{\bar{n}}^2=1\text{ and }{\sigma }_i^2=1\text{ for }1\le i\le n;$$

$${\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i\text{ for }1\le i,j\le n,|i-j|\ge 2;$$

$${\sigma }_i{\sigma }_{\bar{1}}={\sigma }_{\bar{1}}{\sigma }_i\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{ for }i\ne 2;$$

$${\sigma }_i{\sigma }_{\bar{n}}={\sigma }_{\bar{n}}{\sigma }_i\text{ for }i\ne n-1;$$

$${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}\text{ for }1\le i\le n-1;$$

$${\sigma }_2{\sigma }_{\bar{1}}{\sigma }_2={\sigma }_{\bar{1}}{\sigma }_2{\sigma }_{\bar{1}};$$

$${\sigma }_{n-1}{\sigma }_{\bar{n}}{\sigma }_{n-1}={\sigma }_{\bar{n}}{\sigma }_{n-1}{\sigma }_{\bar{n}};$$

$$t_{n+1}^2=1,{{\sigma }_{\bar{1}}}^2=1\text{ and }{\sigma }_i^2=1\text{ for }1\le i\le n;$$

$${\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i\text{ for }1\le i,j\le n,|i-j|\ge 2;$$

$${\sigma }_i{t}_{n+1}=t_{n+1}{\sigma }_i\text{ for }1\le i<n;$$

$${\sigma }_{\bar{1}}{t}_{n+1}=t_{n+1}{\sigma }_{\bar{1}};$$

$${\sigma }_i{\sigma }_{\bar{1}}={\sigma }_{\bar{1}}{\sigma }_i\text{ for }i=1\text{ or }3\le i;$$

$${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}\text{ for }1\le i\le n-1;$$

$${\sigma }_{\bar{1}}{\sigma }_2{\sigma }_{\bar{1}}={\sigma }_2{\sigma }_{\bar{1}}{\sigma }_2;$$

$${\sigma }_n{t}_{n+1}{\sigma }_n{t}_{n+1}=t_{n+1}{\sigma }_n{t}_{n+1}{\sigma }_n.$$

$$t_{n+1}^2=1\text{ and }{\sigma }_i^2=1\text{ for }0\le i\le n;$$

$${\sigma }_i{\sigma }_j={\sigma }_j{\sigma }_i\text{ for }0\le i,j\le n,|i-j|\ge 2;$$

$${\sigma }_i{t}_{n+1}=t_{n+1}{\sigma }_i\text{ for }0\le i<n;$$

$${\sigma }_i{\sigma }_{i+1}{\sigma }_i={\sigma }_{i+1}{\sigma }_i{\sigma }_{i+1}\text{ for }1\le i\le n-1;$$

$${\sigma }_0{\sigma }_1{\sigma }_0{\sigma }_1={\sigma }_1{\sigma }_0{\sigma }_1{\sigma }_0;$$

$${\sigma }_n{t}_{n+1}{\sigma }_n{t}_{n+1}=t_{n+1}{\sigma }_n{t}_{n+1}{\sigma }_n.$$

$$F_n:W({\tilde{C}}_n)$$

$${\sigma }_i$$

$$t_n$$

$$L_n:W({\tilde{B}}_n)$$

$${\sigma }_i$$

$${\sigma }_{\bar{1}}$$

$$t_n$$

$$G_n:W({\tilde{D}}_n)$$

$${\sigma }_i$$

$${\sigma }_{\bar{1}}$$

$${\sigma }_{\overline{n-1}}$$

$$t_{n+1}{\sigma }_n{t}_{n+1}{\sigma }_n={\sigma }_n{t}_{n+1}{\sigma }_n{t}_{n+1}$$

$$x{\sigma }_n{x}^{-1}={\sigma }_n{t}_{n+1}{\sigma }_n{t}_{n+1}{\sigma }_n=({\sigma }_n{t}_{n+1}{\sigma }_n){\sigma }_n({\sigma }_n{t}_{n+1}{\sigma }_n)=I(t_n){\sigma }_{n}I(t_n).$$

$$h_j=(s_1\cdots s_{j-1})s_j(s_1\cdots s_{j-1}{)}^{-1}\text{ for }1\le j\le r.$$

$$w=u_0{t}_n{u}_1\dots u_{s-1}{t}_n{u}_s$$

$$(\ast )I(w)=u_0{\sigma }_n{t}_{n+1}{\sigma }_n{u}_1\dots u_{s-1}{\sigma }_n{t}_{n+1}{\sigma }_n{u}_s.$$

$$\Phi (I(w))=I(\Phi (w))⨆T_1⨆T_2$$

$$MCard(\Phi (I(w)))=Card(\Phi (I(w)))=l(w)+2L(w).$$

$$F_n:W({\tilde{C}}_n)$$

$$\text{ and }{L}_n:W({\tilde{B}}_n)$$

$$l(F_n(v))$$

$$l(L_n(w))$$

$$\begin{array}{rl}& g_s{g}_w=g_{sw}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{leavevmode}\text{ if }s\notin \mathcal{L}(w),\\ & g_s{g}_w=q{g}_{sw}+(q-1)g_w\text{leavevmode}\text{leavevmode}\text{ if }s\in \mathcal{L}(w).\end{array}$$

Full text

IMAFI, Universidad de Talca.

Supported by FONDECYT grant number 3170544.

1. Introduction

A great deal of this work is a continuation of [3] and [4], that is: giving a reduced expression that is a normal form for any fully commutative element in $\tilde{B}$-type and $\tilde{D}$-type Coxeter groups, viewing them as objects in infinite towers and using this view to show the injectivity of the associated towers of affine Temperley-Lieb algebras. In the above mentioned works $\tilde{A}$-type and $\tilde{C}$-type were treated.

Let $(W,S)$ be a Coxeter system. We say that w in $W$ is fully commutative if any reduced expression for $w$ can be obtained from any other using only commutation relations among the members of the set $S$. Call the set of such elements $W^{c}$.

Enumerating fully commutative elements by a normal form is an interesting subject in its own (see Stembridge [11] for the three infinite families of finite Coxeter groups), and also has important applications and consequences. For the infinite affine families, the author treated the $\tilde{A}$-type for a question coming from braid group theory, and, encouraged by a question of Luis Paris, went on, in the same spirit, to treat the three other affine cases.

First of all it has applications to the study of Temperley-Lieb algebras – which are quotients of Hecke algebras – since fully commutative elements in a Coxeter group index at least two known bases of the corresponding Temperley-Lieb algebra. In particular, as a consequence of the normal form for affine types $\tilde{A}$ to $\tilde{D}$, we can prove the injectivity of associated towers of Temperley-Lieb algebras, by producing injective towers of the sets of fully commutative elements. It can also be a decisive tool in the study of traces, particularly traces of Markov type, on those towers of Temperley-Lieb algebras, as was done in [2].

Other than that, by providing accessible computations of some Kazhdan-Lusztig polynomials, as was done in [8], it can give a way to attack the study of the 0-1 conjecture, or of a fully commutative version of this conjecture since the general formulation has proved wrong [10].

In yet another direction, the author in a forthcoming work is giving an explicit Coxeter-length generating function for fully commutative elements as a direct application of this enumeration, moreover an affine-length generating function (see below). This gives a clear idea about the growth of the infinite-dimensional affine Temperley-Lieb algebras.

Towers are to be defined here on four levels: Coxeter groups and Hecke algebras, fully commutative elements and Temperley-Lieb algebras; and in two affine types $\tilde{B}$ and $\tilde{D}$. We prove the injectivity on the Coxeter group level, a crucial step towards injectivity on the Hecke level, which is more technical though.

We gave proofs of injectivity of the tower of affine Hecke algebras over the field $\mathds{Q}[q,q^{-1}]$ in [1, Proposition 4.3.3] for type $\tilde{A}$ and in [4, Proposition 3.3] for type $\tilde{C}$, using specialization at $q=1$, that maps Hecke algebras onto group algebras. We give here a proof of injectivity for type $\tilde{B}$ which is independent of the ring of definition and goes as well for type $\tilde{C}$, hence extending the result in [4]. But for $\tilde{D}$-type the proof keeps similar to type $\tilde{A}$ and needs the same condition on the ground ring.

Let us pause for this. An element in a Coxeter group is called rigid if it has only one reduced expression. In the monomorphisms $W(\tilde{B}_{n})\hookrightarrow W(\tilde{B}_{n+1})$ and $W(\tilde{C}_{n})\hookrightarrow W(\tilde{C}_{n+1})$ defined below, the Coxeter generators of the first group map to rigid elements of the second; however this does not hold for $W(\tilde{A}_{n})\hookrightarrow W(\tilde{A}_{n+1})$ nor $W(\tilde{D}_{n})\hookrightarrow W(\tilde{D}_{n+1})$. The author believes that our proof for type $\tilde{B}$ is generic for any reflexion subgroup of a Coxeter group whenever the reflexions are rigid!

Precisely, recall from [6] that a subgroup $Y$ of a Coxeter group $(W,S)$ generated by reflexions is itself a Coxeter group $(Y,S_{Y})$ with a canonical set of reflexions $S_{Y}$. Paolo Sentinelli has recently asked whether, in this situation, the corresponding Hecke algebras embed accordingly. Actually the existence of such a morphism is not guaranteed. For instance, there is no morphism of Hecke algebras attached to the monomorphism $\beta:W(\tilde{B}_{n})\hookrightarrow W(\tilde{C}_{n})$ in §2 (see Remark 2.3). The author believes the following: if there exists a corresponding homomorphism of Hecke algebras, then it is injective, and in the case where the elements of $S_{Y}$ are rigid in $(W,S)$, that it can be proved by elaborating on the present proof in type $\tilde{B}$.

The classification of fully commutative elements and the proof of injectivity of the towers of Temperley-Lieb algebras in the four affine types are based on the notion of "affine length", this is to be explained briefly as follows: we choose to see any affine Coxeter group as an "affinization" of a finite Coxeter group in the obvious way (but for the case of $\tilde{B}$-type which can be viewed as an extension of either $B$-type or $D$-type, we choose the latter for technical reasons). This "affinization" is simply adding a simple reflexion, say $a$, as follows:

We thus define the affine length of any fully commutative element to be the number of occurrences of $a$ in a (hence any) reduced expression for this fully commutative element. More general: Let $(W,S)$ be any Coxeter group; for any $w$ in $W^{c}$, let $a$ be any element of $S$, then the number of occurrences of $a$ in any reduced expression of $w$ is independent of the reduced expression, by Matsumoto’s theorem.

Moreover, by distinguishing this $a$ we see simply and directly the construction of towers of each type, based on the morphisms in §2.2 that map, say, $a_{n}$ to a suitable conjugate of $a_{n+1}$ while fixing the other generators. In types $\tilde{B}$ and $\tilde{C}$, these morphisms $L_{n}$ and $F_{n}$ have a remarkable property: the image of $a_{n}$ is rigid, as mentioned previously. Furthermore, for those two types, the braid relation involving $a$ has even length so that the affine length can be defined for any element of the Coxeter group. Eventually we prove that these two morphisms preserve reduced expressions (Corollary 2.7), a key argument for proving the injectivity of the tower of Hecke algebras.

This work is to be presented as follows: in the section 2 we give our notation for the presentation of the Coxeter groups we will be working on, including the "affinization" explained above. We define morphisms between them that give rise to injective towers of affine Coxeter groups of a given type (Corollary 2.2). With a closer examination we prove that in types $\tilde{B}$ and $\tilde{C}$ those morphisms preserve reduced expressions (Theorem 2.6 and Corollary 2.7).

Then in section 3 we define the corresponding towers of Hecke algebras. We give a proof of their faithfulness in types $\tilde{B}$ and $\tilde{C}$ over any ring (i.e. commutative ring with identity) where $q$ is invertible. The proof relies on Corollary 2.7 that is not available in type $\tilde{D}$. For type $\tilde{D}$ the faithfulness of the tower of Hecke algebras can be proven on $\mathds{Q}[q,q^{-1}]$ as was done before in [1].

In section 4 we recall the normal form given by Stembridge for $D$-type fully commutative elements [11, Theorem 10.1] and present our normal form for $\tilde{B}$-type and $\tilde{D}$-type fully commutative elements. In both cases we distinguish between two types of elements of affine length at least 2, that we call simply first type elements and second type elements. Elements of affine length 1 cannot be attached to these two categories without ambiguity so we keep them apart.

We show in section 5 that the normal form of a fully commutative element in $W(\tilde{B}_{n})$ (resp.$W(\tilde{D}_{n})$ can be transformed into the normal form of a fully commutative element in $W(\tilde{B}_{n+1})$ (resp.$W(\tilde{D}_{n+1})$ in two different ways, that actually coincide for first type elements and elements of affine length one. We obtain two injective maps $I$ and $J$ that play an essential part in section 6, where we study the morphisms of Temperley-Lieb algebras $Q_{n}:TL\tilde{B}_{n}(q)\longrightarrow TL\tilde{B}_{n+1}(q)$ and $P_{n}:TL\tilde{D}_{n}(q)\longrightarrow TL\tilde{D}_{n+1}(q)$ induced from the morphisms of Hecke algebras of section 3.

Indeed, we can describe the image of a basis element indexed by a fully commutative element $w$ as a linear combination of basis elements in which the terms of highest affine length and highest Coxeter length are indexed by $I(w)$ and $J(w)$ (Lemma 6.2 and Proposition 6.3; Proposition 6.6) or by $I(w)$ and some $I(\bar{w})$ when $w$ has affine length 1 in type $\tilde{D}$ (Lemma 6.7). This implies the faithfulness of the towers of Temperley-Lieb algebras: Theorem 6.4 and Theorem 6.8.

2. Faithful towers of Coxeter groups

Let $(W(\Gamma),S)$ be a Coxeter system with associated Coxeter diagram $\Gamma$. Let $w\in W(\Gamma)$ or simply $W$. We denote by $l(w)$ the length of a (any) reduced expression of $w$. We define $\mathscr{L}(w)$ to be the set of $s\in S$ such that $l(sw)<l(w)$, in other terms $s$ appears at the left edge of some reduced expression of $w$. We define $\mathscr{R}(w)$ similarly, on the right.

2.1. Presentations of $D_{n+1}$, $\tilde{D}_{n+1}$, $\tilde{B}_{n+1}$, $\tilde{C}_{n+1}$

For $n\geq 3$ consider the $D$-type Coxeter group with $n+1$ generators $W(D_{n+1})$, with the following Coxeter diagram:

Now let $W(\tilde{D}_{n+1})$ be the affine Coxeter group of $\tilde{D}$-type with $n+2$ generators in which $W(D_{n+1})$ could be seen a parabolic subgroup in two ways. We make our choice by presenting $W(\tilde{D}_{n+1})$ with the following Coxeter diagram:

In other words the group $W(\tilde{D}_{n+1})$ has a presentation given by the set of generators $\{\sigma_{1},\sigma_{\bar{1}}\dots,\sigma_{n-1},\sigma_{\bar{n}},\sigma_{n}\}$ and the relations:

[TABLE]

Now let $W(\tilde{B}_{n+1})$ be the affine Coxeter group of $\tilde{B}$-type with $n+2$ generators in which $W(D_{n+1})$ is naturally a parabolic subgroup, as seen in the following Coxeter diagram:

In other words the group $W(\tilde{B}_{n+1})$ has a presentation given by the set of generators $\{\sigma_{1},\sigma_{\bar{1}},\dots,\sigma_{n},t_{n+1}\}$ and the relations:

[TABLE]

Now we consider the $\tilde{C}$-type Coxeter group with $n+2$ generators $W(\tilde{C}_{n+1})$, having as Coxeter diagram:

In other terms we generate $W(\tilde{C}_{n+1})$ by $\{\sigma_{0},\sigma_{1},\dots,\sigma_{n},t_{n+1}\}$ subject to the following relations:

[TABLE]

2.2. Faithfulness of towers of $\tilde{B}$-type and $\tilde{D}$-type affine Coxeter groups

In [4, Corollary 2.2] the author has defined a faithful tower of $\tilde{C}$-type Coxeter groups by defining the following homomorphism for $n\geq 2$ and proving its injectivity:

[TABLE]

We define in a similar way the two following homomorphisms:

[TABLE]

[TABLE]

In [9, §1.13] we see that for $n>2$, in $W(\tilde{C}_{n+1})$ the subgroup generated by $\{t\sigma_{1}t,\sigma_{1},\sigma_{2},\dots,\sigma_{n},t_{n+1}\}$ is exactly $W(\tilde{B}_{n+1})$, and in the later the subgroup generated by $\{t\sigma_{1}t,\sigma_{1},\sigma_{2},\dots,\sigma_{n},t_{n+1}\sigma_{n}t_{n+1}\}$ is indeed $W(\tilde{D}_{n+1})$. It is convenient to denote respectively $\beta$ and $\delta_{n+1}$ those inclusion maps (that is, $\beta(\sigma_{\bar{1}})=t\sigma_{1}t$ and $\delta_{n+1}(\sigma_{\bar{n}})=t_{n+1}\sigma_{n}t_{n+1}$ , in such a way that we have the following lemma:

Lemma 2.1**.**

The diagram commutes for any $n>2$

{W(\tilde{C}_{n})}$${W(\tilde{C}_{n+1})}$${W(\tilde{B}_{n})}$${W(\tilde{B}_{n+1})}$${W(\tilde{D}_{n})}$${W(\tilde{D}_{n+1})}$$\beta$$\delta_{n}$$F_{n}$$L_{n}$$G_{n}$$\beta$$\delta_{n+1}

Corollary 2.2**.**

$L_{n}$* and $G_{n}$ are injections.

Remark 2.3**.**

The horizontal morphisms in the previous lemma, namely $F_{n}$, $L_{n}$ and $G_{n}$, can be defined on the corresponding braid groups (by mapping $t_{n}$ to ${\sigma_{n}}t_{n+1}{\sigma_{n}}^{-1}$ for $F_{n}$ and $L_{n}$, or mapping $\sigma_{\bar{n-1}}$ to $\sigma_{n}\sigma_{n-1}\sigma_{\bar{n}}\sigma_{n-1}^{-1}\sigma_{n}^{-1}$ for $G_{n}$). They thus define morphisms of algebra on the braid group algebras and, after taking quotients, on the Hecke algebras studied in the next section. This is not the case for $\delta_{n}$ or $\beta$.

2.3. Further properties of the towers in type $\tilde{B}$ and $\tilde{C}$

We now remark that, in the groups $W(\tilde{B}_{n+1})$ and $W(\tilde{C}_{n+1})$, the only braid relation involving $t_{n+1}$ (apart from commutation relations) is

[TABLE]

where the number of occurrences of $t_{n+1}$ is the same on both sides. It follows (recall [5, §1.5 Proposition 5]) that the number of times $t_{n+1}$ occurs in a reduced expression of an element of $W(\tilde{B}_{n+1})$ or $W(\tilde{C}_{n+1})$ does not depend of this reduced expression.

Definition 2.4**.**

Let $u\in W(\tilde{B}_{n+1})$ or $W(\tilde{C}_{n+1})$. We define the affine length of $u$ to be the number of times $t_{n+1}$ occurs in a (any) reduced expression of $u$. We denote it by $L(u)$.

In the group $W(\tilde{D}_{n+1})$, the braid relation $\sigma_{n-1}\sigma_{\bar{n}}\sigma_{n-1}=\sigma_{\bar{n}}\sigma_{n-1}\sigma_{\bar{n}}$ prevents us from giving the same definition, except for fully commutative elements as we will see below (Definition 4.5).

We will now examine more closely the monomorphisms $F_{n}:W(\tilde{C}_{n})\longrightarrow W(\tilde{C}_{n+1})$ and $L_{n}:W(\tilde{B}_{n})\longrightarrow W(\tilde{B}_{n+1})$ and show that they preserve the affine length and transform the Coxeter length of some $w$ into $l(w)+2L(w)$. In other words, when substituting ${\sigma_{n}}t_{n+1}{\sigma_{n}}$ to $t_{n}$ in a reduced expression for $w$, we obtain a reduced expression for the image of $w$.

We let in this subsection $W_{n}$ be $W(\tilde{C}_{n})$ or $W(\tilde{B}_{n})$, with set of generators $S_{n}=\Sigma\cup\{t_{n}\}$ while $W_{n+1}$ has $S_{n+1}=\Sigma\cup\{\sigma_{n},t_{n+1}\}$ as set of generators. We let $I$ be either $F_{n}$ or $L_{n}$, i.e. $I$ is the monomorphism from $W_{n}$ to $W_{n+1}$ that is the identity on generators in $\Sigma$ and maps $t_{n}$ to $\sigma_{n}t_{n+1}\sigma_{n}$. We also let $W_{\Sigma}$ be the subgroup of $W_{n}$ generated by $\Sigma$.

Lemma 2.5**.**

*Let $x$ be in $I(W_{n})$ then:

*(a) $t_{n+1}x=xt_{n+1}$ ;

*(b) If $\sigma_{n}x=x\sigma_{n}$ then $I(t_{n})x=xI(t_{n})$;

*(c) If $xt_{n+1}\sigma_{n}=\sigma_{n}t_{n+1}x$ then $I(t_{n})x=xI(t_{n})$;

Proof.

(a) Indeed $t_{n+1}$ commutes with generators in $\Sigma$ and the braid relation (4) expresses commutation with $T(t_{n})$. (b) follows immediately.

Using (a) the assumption in (c) can be written $x\sigma_{n}x^{-1}=t_{n+1}\sigma_{n}t_{n+1}$, hence by (4):

[TABLE]

So $I(t_{n})x$ commutes with $\sigma_{n}$, hence with $I(t_{n})$ by (b), which gives

$I(t_{n})I(t_{n})x=I(t_{n})xI(t_{n})$, that is $I(t_{n})x=xI(t_{n})$. ∎

We now remember [5, Ch. IV, §1.4]. Given a Coxeter system $(W,S)$, we attach to any finite sequence $\mathbf{s}=(s_{1},\cdots,s_{r})$ of elements in $S$, the multiset $\Phi(\mathbf{s})=\{\{h_{1},\cdots,h_{r}\}\}$ of elements in $W$ defined by:

[TABLE]

We write $\operatorname{MCard}\Phi(\mathbf{s})$ for the multi-cardinal of $\Phi(\mathbf{s})$, i.e.the number of elements in the multiset $\Phi(\mathbf{s})$, namely $r$ here, and $\operatorname{Card}\Phi(\mathbf{s})$ for the cardinality of the set underlying $\Phi(\mathbf{s})$. We know from loc.cit. Lemma 2 that:

the product $s_{1}\cdots s_{r}$ is a reduced expression of the element $s_{1}\cdots s_{r}$ in $W$ if and only if $\ \operatorname{MCard}\Phi(\mathbf{s})=\operatorname{Card}\Phi(\mathbf{s})$, that is, all elements in the multiset are distinct.

We now let $w\in W_{n}$ and fix a reduced decomposition of $w$ as follows :

[TABLE]

where $u_{0},\cdots,u_{s}$ are fixed reduced decompositions of elements in $W_{\Sigma}$. In particular the affine length of $w$ is $s$. The image $I(w)$ in $W_{n+1}$ is

[TABLE]

We use this fixed expression of $I(w)$ to study what we call, by abuse of language, $\Phi(I(w))$. We can describe this multiset as a disjoint union of multisets

[TABLE]

as follows. We develop the expressions of $u_{0},\dots,u_{s}$ so that $(\ast)$ reads $I(w)=s_{1}\cdots s_{r}$, then we have three possibilities for an element $h_{i}=(s_{1}\cdots s_{i-1})\ s_{i}\ (s_{1}\cdots s_{i-1})^{-1}$ of $\Phi(I(w))$:

1. (1)

$s_{i}\in\Sigma$ or $s_{i}=t_{n+1}$; then $h_{i}=xXx^{-1}$ where $x$ and $X$ are in $I(W_{n})$. Those elements make up exactly the multiset $I(\Phi(w))$. 2. (2)

$s_{i}=\sigma_{n}$ and $s_{i+1}=t_{n+1}$; then $h_{i}=x\sigma_{n}x^{-1}$ with $x$ in $I(W_{n})$. Those elements make up the multiset $T_{1}$. 3. (3)

$s_{i}=\sigma_{n}$ and $s_{i-1}=t_{n+1}$; then $h_{i}=x\sigma_{n}t_{n+1}\sigma_{n}t_{n+1}\sigma_{n}x^{-1}=xt_{n+1}\sigma_{n}t_{n+1}x^{-1}$ with $x$ in $I(W_{n})$. Those elements make up the multiset $T_{2}$.

We have $\operatorname{MCard}(T_{1})=\operatorname{MCard}(T_{2})=L(w)$ and $\operatorname{MCard}(I(\Phi(w)))=l(w)$, since $w$ is given in a reduced expression (we will anyway see it in (1) of the following theorem). Hence the multi-cardinal of $\Phi(I(w))$ is $l(w)+2L(w)$. If the elements of $\Phi(I(w))$ are all distinct, then it is the cardinal.

Theorem 2.6**.**

Let $w\in W_{n}$ and take the above conventions. Then

[TABLE]

*In particular, substituting \ {\sigma_{n}}t_{n+1}{\sigma_{n}}\ to \ t_{n}\ in a reduced expression for $w$ produces a reduced expression for the image $I(w)$ of $w$.

Proof.

Take $h_{i}$ and $h_{j}$ in $\Phi(I(w))$ with $i\neq j$ and suppose that $h_{i}=h_{j}$. We have six cases.

(1) $h_{i}$ and $h_{j}$ are in $I(\Phi(w))$: contradicts that $w$ is given in its reduced expression and $I$ is injective.

(2) $h_{i}$ and $h_{j}$ are in $T_{1}$, gives (b) in the above lemma, thus contradicts that $w$ is given in its reduced expression.

(3) $h_{i}$ and $h_{j}$ are in $T_{2}$, gives (b) in the above lemma, thus contradicts that $w$ is given in its reduced expression.

(4) $h_{i}$ is in $I(\Phi(w))$ and $h_{j}$ is in $T_{1}$, gives $\sigma_{n}\in I(W_{n})$, hence commutes with $t_{n+1}$ by (a) in the above lemma, contradiction.

(5) $h_{i}$ is in $I(\Phi(w))$ and $h_{j}$ is in $T_{2}$, gives $t_{n+1}\sigma_{n}t_{n+1}\in I(W_{n})$, hence commutes with $t_{n+1}$ by (a) in the above lemma, contradiction.

(6) $h_{i}$ is in $T_{1}$ and $h_{j}$ is in $T_{2}$, gives (c) in the above lemma, thus contradicts that $w$ is given in its reduced expression. ∎

Corollary 2.7**.**

The monomorphisms

[TABLE]

satisfy, for any $v\in W(\tilde{C}_{n})$ and for any $w\in W(\tilde{B}_{n})$ :

[TABLE]

3. The towers of Hecke algebras

Let for the moment $K$ be an arbitrary commutative ring with identity; we mean by algebra in what follows $K$-algebra. We recall [5, Ch. IV §2 Ex. 23] that for a given Coxeter graph $\Gamma$ and a corresponding Coxeter system $(W(\Gamma),S)$, there is a unique algebra structure on the free $K$-module with basis $\left\{g_{w}\ |\ w\in W(\Gamma)\right\}$ satisfying, for $s\in S$ and a given $q\in K$:

[TABLE]

We denote this algebra by $H\Gamma(q)$ and call it the the $\Gamma$-type Hecke algebra. This algebra has a presentation (loc.cit.) given by generators $\left\{g_{s}\ |\ s\in S\right\}$ and relations

[TABLE]

We assume in what follows that $q$ is invertible in $K$. In this case the first defining relation above implies that $g_{s}$, for $s\in S$, is invertible with inverse

[TABLE]

We consider the Hecke algebras $H\tilde{C}_{n}(q)$, of type $\tilde{C_{n}}$, $H\tilde{B}_{n}(q)$, of type $\tilde{B_{n}}$, and $H\tilde{D}_{n}(q)$, of type $\tilde{D_{n}}$, which correspond respectively to the Coxeter groups $W(\tilde{C}_{n})$, $W(\tilde{B}_{n})$ and $W(\tilde{D}_{n})$. The morphisms $F_{n}$, $L_{n}$ and $G_{n}$ of §2 have a counterpart in the setting of Hecke algebras, as follows from Remark 2.3 or as can be checked directly, namely the following morphisms of algebras (where we write carefully $e_{w}$ for the basis elements of the algebras in rank $n$, to be reminded of the possible lack of injectivity):

[TABLE]

[TABLE]

[TABLE]

When $K$ is the ring $\mathbf{Q}[q,q^{-1}]$ of Laurent polynomials in $q$ with rational coefficients, we proved in [4, Proposition 3.3] that the morphism $R_{n}$ is injective, hence producing a faithful tower of Heche algebras of type $\tilde{C}$. Actually for types $\tilde{C}$ and $\tilde{B}$ we prove below injectivity for any ground ring $K$. For type $\tilde{D}$ we keep to the special case $K=\mathbf{Q}[q,q^{-1}]$ but we will skip the details of the proof, similar to the proof in loc.cit.. In any case, injectivity at the level of Hecke algebras is not needed to prove injectivity for towers of Temperley-Lieb algebras in §6.

3.1. Faithfulness of towers of $\tilde{B}$-type and $\tilde{C}$-type Hecke algebras

For types $\tilde{B}$ and $\tilde{C}$ the key is the following Proposition:

Proposition 3.1**.**

Let $w$ be any element in $W(\tilde{C}_{n})$. Then:

[TABLE]

*where $A_{w}$ belongs to $q^{\mathds{Z}}$ and the $\lambda_{x}$ belong to $K$.

*The same holds for $w\in W(\tilde{B}_{n})$, replacing $R_{n}$ by $Q_{n}$ and $F_{n}$ by $L_{n}$.

Proof.

This is a direct consequence of Corollary 2.7. To obtain it we use definitions (6) of $R_{n}$ and (7) of $Q_{n}$ and develop the images $R_{n}(e_{w})$ and $Q_{n}(e_{w})$ with the expression (5) for the inverse of $g_{\sigma_{n}}$. ∎

Theorem 3.2**.**

Let $K$ be a ring and $q$ be invertible in $K$. The towers of affine Hecke algebras:

[TABLE]

*are towers of faithful arrows.

Proof.

We write the proof in case $\tilde{C}$, case $\tilde{B}$ is identical. Assuming a non trivial dependence relation

[TABLE]

we let $m=\max\{L(w)\ |\ w\in W^{c}(\tilde{C}_{n})\text{ and }\lambda_{w}\neq 0\}.$

In the development of $(\ast)$ on the basis $(g_{x})_{x\in W(\tilde{C}_{n+1}}$, the terms $g_{x}$ with non-zero coefficient have $L(x)\leq m$ and, among those, the ones with $L(x)=m$ come exclusively from $\sum_{L(w)=m}\lambda_{w}R_{n}(e_{w})$. So among those again, using Proposition 3.1, the ones with maximal Coxeter length are the $\lambda_{w}A_{w}g_{F_{n}(w)}$ where $L(w)=m$ and $l(w)=a$, with $a=\max\{l(w)\ |\ L(w)=m\text{ and }\lambda_{w}\neq 0\}$. But they are linearly independent, a contradiction. ∎

3.2. Faithfulness of towers of $\tilde{D}$-type Hecke algebras

For type $\tilde{D}$, as in type $\tilde{A}$, where the proof of injectivity was given in [1, Proposition 4.3.3], the preceeding proof cannot be used because the monomorphism $G_{n}$ defined in Corollary 2.2 does not transform the length in the linear way $F_{n}$ and $L_{n}$ do. The proof of [1, Proposition 4.3.3] or [4, Proposition 3.3] have to be imitated to get the following Proposition. Since we expect injectivity to be holding more generally, we don’t include the proof.

Proposition 3.3**.**

Let $K=\mathbf{Q}[q,q^{-1}]$. The homomorphism of algebras

[TABLE]

*defined in (8) is an injection.

4. Full commutativity and normal forms

4.1. Fully commutative elements

In a given Coxeter group $(W,S)$, we know that, from a given reduced expression of an element $w$, we can arrive to any other reduced expression of $w$ only by applying braid relations [5, §1.5 Proposition 5]. Among these relations there are commutation relations: those that correspond to generators $t$ and $s$ in $S$ such that $st$ has order $2$.

Definition 4.1**.**

*Let $\Gamma$ be a Coxeter diagram. Elements of $W(\Gamma$) for which one can pass from any reduced expression to any other one only by applying commutation relations are called fully commutative elements. We denote by $W^{c}(\Gamma)$, or simply $W^{c}$, the set of fully commutative elements in $W=W(\Gamma)$.

We now consider $W(D_{n+1})$ generated by $\{\sigma_{1},\sigma_{\bar{1}},\dots,\sigma_{n}\}$. The set $W^{c}(D_{n+1})$ is described by Stembridge in [11]. We use the notation there and the same assumption that the subword $\sigma_{\bar{1}}\sigma_{1}$ does not appear (we see it as $\sigma_{1}\sigma_{\bar{1}}$ for the sake of unicity). For integers $j\geq i\geq 2$ and $k\geq 1$ let:

[TABLE]

so that $\langle-1,1]=\sigma_{\bar{1}}$ and $\langle 0,1]=\sigma_{1}\sigma_{\bar{1}}$. We also let, for convenience and later use: $\langle n+1,n]=1$. Then every element of $W(D_{n+1})$ has a unique reduced expression of the form $\langle m_{1},n_{1}]\langle m_{2},n_{2}]\dots\langle m_{r},n_{r}]$ with $n\geq n_{1}>n_{2}>\dots n_{r}\geq 1$ and $|m_{i}|\leq n_{i}$ for $1\leq i\leq r$ (loc.cit. p.1310). Now:

Theorem 4.2**.**

[11, Theorem 10.1]* $W^{c}(D_{n+1})$ is the set of elements with a reduced expression of the form*

[TABLE]

with $n\geq n_{1}>n_{2}>\dots n_{r}\geq 1$ and $|m_{i}|\leq n_{i}$ for $1\leq i\leq r$, where the occurrences of $1$ and $-1$ alternate and either

1. (1)

$m_{1}>\dots>m_{s}>|m_{s+1}|=\dots=|m_{r}|=1$* for some $s\leq r$, or* 2. (2)

$m_{1}>\dots>m_{r-1}>-m_{r}\geq 0$*, $m_{r-1}>1$, and $m_{r}\neq-1$. *

We provide as an example in Appendix A the list of elements of $W^{c}(D_{4})$.

We remark that if $r>1$, then $\langle m_{2},n_{2}]\dots\langle m_{s},n_{s}]$ in (9) above belongs to $W^{c}(D_{n})$. From now on we will mean by "the form (9)" the form of a fully commutative element unless we mention otherwise. We notice that if $\sigma_{n}$ appears in form (9) above, then it appears necessarily in the first term $\langle m_{1},n_{1}]$ and either it appears only once and we have $n=n_{1}\neq-m_{1}$, or it appears exactly twice and we have $n=n_{1}=-m_{1}$ and $r=1$. Now $\sigma_{n-1}$ can only appear in the two first terms $\langle m_{1},n_{1}]\langle m_{2},n_{2}]$ and this happens if and only if $n_{1}\geq n-1$. If $\sigma_{n-1}$ appears more than once, then in form (9) either $n_{2}=n-1$ and $n_{1}=n$, in which case the constraints of Theorem 4.2 show that $\sigma_{n-1}$ appears at most twice (for if $m_{2}=-(n-1)$ then $m_{1}=n$); or $n_{2}<n-1\leq n_{1}$ and $m_{1}\leq-(n-1)$ so $r=1$.

Definition 4.3**.**

An element $u$ in $W^{c}(D_{n+1})$ is called $\tilde{B}$-extremal if $\sigma_{n}$ appears in a (any) reduced expression for $u$. In this case $u$ can be written in form (9) with $n_{1}=n$ and either $u=\langle-n,n]$, or $m_{1}>-n$ and $\sigma_{n}$ appears only once.

Definition 4.4**.**

An element $u$ in $W^{c}(D_{n+1})$ is called $\tilde{D}$-extremal if $\sigma_{n-1}$ appears twice in a (any) reduced expression for $u$. In this case $u$ can be written in exactly one of the four following forms:

• •

$\langle-(n-1),n-1]$* ;*

• •

$\langle-(n-1),n]$* ;*

• •

$\langle n,n]\langle-(n-1),n-1]$* ;*

• •

\langle m_{1},n]\langle m_{2},n-1]\dots\langle m_{r},n_{r}]\* with $r\geq 2$ and $m_{1}\leq n-1$ in addition to the conditions of Theorem 4.2.*

We defined previously the affine length of an element of $W(\tilde{B}_{n+1})$ (Definition 2.4), which we couldn’t do on the full group $W(\tilde{D}_{n+1})$ on account of the braid relation

[TABLE]

Since the words here cannot be subwords of any reduced expression of an element of $W^{c}(\tilde{D}_{n+1})$, it follows (recall [5, §1.5 Proposition 5]) that the number of times $\sigma_{\bar{n}}$ occurs in a reduced expression of an element of $W^{c}(\tilde{D}_{n+1})$ does not depend of this reduced expression.

Definition 4.5**.**

*Let $u\in W^{c}(\tilde{D}_{n+1})$. We define the affine length of $u$ to be the number of times $\sigma_{\bar{n}}$ occurs in a (any) reduced expression of $u$. We denote it by $L(u)$.

4.2. Reduced expressions of elements of $W^{c}(\tilde{B}_{n+1})$

Lemma 4.6**.**

Let $w$ be a fully commutative element in $W(\tilde{B}_{n+1})$ with $L(w)=m\geq 2$ for $n\geq 3$ . Fix a reduced expression of $w$ as follows:

[TABLE]

with $u_{i}$, for $1\leq i\leq m+1$, a reduced expression of an element in $W^{c}(D_{n+1})$. Then $u_{2},\dots,u_{m}$ are $\tilde{B}$-extremal elements and there is a reduced expression of $w$ of the form:

[TABLE]

where $v_{m+1}\in W^{c}(D_{n+1})$ and one of the following holds:

1. (1)

$n+1\geq i_{1}>\dots>i_{s}>|i_{s+1}|=\dots=|i_{m}|=1$* for some $s\leq m$, and the occurrences of $1$ and $-1$ alternate, or* 2. (2)

$n+1\geq i_{1}>\dots>i_{m-1}>-i_{m}\geq 0$, $i_{m-1}>1$, and $i_{m}\neq-1$, or 3. (3)

$n+1\geq i_{1}\geq i_{2}=\dots=i_{m}=-n$* .*

Furthermore, in case (2) we have $v_{m+1}=1$.

Proof.

Since $t_{n+1}$ commutes with $W(D_{n})$, the fact that the expression is reduced forces $u_{i}$ to be $\tilde{B}$-extremal for $2\leq i\leq m$. We use form (9) for $u_{1}$ and write it as $u_{1}=\langle i_{1},n]x_{1}$, a reduced expression with $x_{1}$ in $W^{c}(D_{n})$ and $-n\leq i_{1}\leq n+1$. Here $x_{1}$ commutes with $t_{n+1}$ hence we get a reduced expression

[TABLE]

Again $x_{1}u_{2}\in W^{c}(B_{n+1})$ has a reduced expression $\langle i_{2},n]x_{2}$ with $-n\leq i_{2}\leq n$ (since $x_{1}u_{2}$ is $\tilde{B}$-extremal) and $x_{2}$ in $W^{c}(D_{n})$, and this $x_{2}$ commutes with $t_{n+1}$ and can be pushed to the right, leading to

[TABLE]

Proceeding from left to right we obtain formally form (10).

We now fix $j$, $1\leq j<m$.

• •

If $i_{j+1}=n$ we must have $i_{j}=n+1$ (and $j=1$) since, with $i_{j}\leq n$, our expression would contain the braid $\sigma_{n}t_{n+1}\sigma_{n}t_{n+1}$, contradicting the full commutativity;

• •

while if $i_{j+1}=-n$, then if $j=1$, any $i_{1}$ is possible, but if $j>1$ we must have $i_{j+1}=i_{j}=-n$ to avoid the braid $t_{n+1}\sigma_{n}t_{n+1}\sigma_{n}$.

• •

If $1<|i_{j+1}|<n$ the reduced expression of $u$ contains the subexpression $\langle i_{j},n]t_{n+1}\sigma_{|i_{j+1}|}=\langle i_{j},n]\sigma_{|i_{j+1}|}t_{n+1}=\dots$ where $\sigma_{|i_{j+1}|}$ can be pushed to the left until we reach, if $i_{j}\leq|i_{j+1}|$ the braid $\sigma_{|i_{j+1}|}\sigma_{|i_{j+1}|+1}\sigma_{|i_{j+1}|}$, again a contradiction to the full commutativity; hence we have $i_{j}>|i_{j+1}|$.

• •

If $|i_{j+1}|=1$ the same argument, pushing $\sigma_{1}$ or $\sigma_{\bar{1}}$ to the left, past $t_{n+1}$ and further, shows that we must have $i_{j}>1$ or $i_{j}=-i_{j+1}$, otherwise we would get the braid $\sigma_{1}\sigma_{2}\sigma_{1}$ or $\sigma_{\bar{1}}\sigma_{2}\sigma_{\bar{1}}$.

• •

Finally if $i_{j+1}=0$, we must have $i_{j}>1$ to avoid the braids $\sigma_{1}\sigma_{2}\sigma_{1}$ and $\sigma_{\bar{1}}\sigma_{2}\sigma_{\bar{1}}$.

Putting together the above rules gives the possibilities announced and a similar argument provides the last assertion. ∎

This lemma leads directly to the classification of fully commutative elements in $W(\tilde{B}_{n+1})$.

Theorem 4.7**.**

Let $n\geq 3$. Let $w\in W^{c}(\tilde{B}_{n+1})$ with $L(w)\geq 2$. Then $w$ can be written in a unique way as a reduced word of one and only one of the following two forms, for non negative integers $p$ and $k$:

**First type: **

[TABLE]

with $-n\leq i\leq n+1$ and $-n\leq f\leq n+1$.

We call such elements quasi-rigid because their reduced expression is unique up to replacing $\sigma_{1}\sigma_{\bar{1}}$ by $\sigma_{\bar{1}}\sigma_{1}$.

**Second type: **

[TABLE]

*with $w_{r}\in W^{c}(D_{n+1})$ and *

• **•: **

if $k>0$:

• **–: **

for $1\leq s\leq k$, we have $j_{s}=(-1)^{a+s}$ for some integer $a$, that is to say $j_{s}=\pm 1$ and $1$ and $-1$ alternate ;

• **–: **

and $w_{r}=\langle(-1)^{a+k+1},r_{1}]\langle(-1)^{a+k+2},r_{2}]\dots\langle(-1)^{a+k+u},r_{u}]$

with $1\leq r_{u}<\dots<r_{1}\leq n$ ;

• **•: **

if $p>0$: \ n+1\geq i_{1}>...>i_{p-1}>|i_{p}|\;

• **•: **

if $p>0$ and $i_{p}\leq 0$: $k=0$, $w_{r}=1$ and $i_{p}\neq-n$ ;

• **•: **

*if $k=0$ and $i_{p}>0$: $w_{r}$ is of form (9) with $|m_{1}|<i_{p}$. *

*The affine length of $w$ of the first (resp. second) type is $k+1$ (resp. $p+k$) and we have $0\leq p\leq n+1$.

Now suppose that $L(w)=1$, then $w$ has a unique reduced expression of the form:

[TABLE]

where $-n\leq i\leq n+1$ and $v\in W^{c}(B_{n+1})$ and:

• •

if $1<i\leq n+1$ then $v$ is of the form (9) such that for $1\leq j\leq r$ either $m_{j}=n-j+1$ or $|m_{j}|<i$;

• •

if $i\leq 0$ and $i\neq-1$ then $v=\langle h,n]^{-1}$ with $-n\leq h\leq n+1$;

• •

if $i=(-1)^{a},$ then either $v=\langle h,n]^{-1}$ for $-n\leq h\leq n+1$, or

$v=\langle(-1)^{a+1},r_{1}]\langle(-1)^{a+2},r_{2}]\dots\langle(-1)^{a+u},r_{u}]$* with $1\leq r_{u}<\dots<r_{1}\leq n$.*

Conversely, every $w$ of the above form is in $W^{c}(\tilde{B}_{n+1})$.

Proof.

We start with an element $w\in W^{c}(\tilde{B}_{n+1})$ with $L(w)=m\geq 2$, written as in (10). Lemma 4.6 provides us with a good part of the Theorem. Indeed we already know that the sequence $i_{1},\dots,i_{m}$ either stops at the first non-positive integer or stabilizes as a sequence of alternating $1$ and $-1$ or as a sequence of $-n$, whence the classification into first and second type. So we just have to deal with the final term $w_{r}$, with the same kind of arguments as in the proof of Lemma 4.6.

For an element of first type, any reduced expression of $w_{r}$ must start with $\sigma_{n}$, actually it must be some right truncation of $\langle-n,n]$, otherwise we would produce a braid and full commutativity would fail. The easiest way to write it is $(\langle f,n])^{-1}$.

For an element of second type with $k>0$, any reduced expression of $w_{r}$ must start with $\sigma_{1}$ if $j_{k}=-1$ or $\sigma_{\bar{1}}$ if $j_{k}=1$. Writing $w_{r}=\langle m_{1},n_{1}]\langle m_{2},n_{2}]\dots\langle m_{u},n_{u}]$ in form (9) we see from the conditions of Theorem 4.2 that indeed $m_{s}=(-1)^{a+k+s}$.

The fact that $w_{r}=1$ for $p>0$ and $i_{p}\leq 0$ is stated in Lemma 4.6. Now if $k=0$ and $i_{p}>0$ we write again $w_{r}$ in form (9) and check that the only additional condition is $|m_{1}|<i_{p}$.

For an element $w=\langle i,n]\ t_{n+1}$, $v\in W^{c}(\tilde{B}_{n+1})$ of affine length $1$, the arguments are similar, using form (9) for $v$. If $i<-1$ or $i=0$, any reduced expression of $v$ must start with $\sigma_{n}$ on the left and must be some right truncation of $\langle-n,n]$, that is, some $\langle h,n]^{-1}$. If $i=\pm 1$, such elements $\langle h,n]^{-1}$ are suitable, as well as those in form (6) that start with $\sigma_{1}$ if $i=-1$ or $\sigma_{\bar{1}}$ if $i=1$, whence the result as previously.

Assume now that $i>1$ and write $v=\langle m_{1},n_{1}]\langle m_{2},n_{2}]\dots\langle m_{u},n_{u}]$ in form (9). If $i=n+1$, there is no further condition on $v$, while if $i\leq n$, any reduced expression of $v$ must start with $\sigma_{n}$ or with $\sigma_{t}$ with $t<i$.

The fact that any element of one of these forms is fully commutative is proven by an easy induction. ∎

We remark that elements of the first type and elements of affine length $1$ of the form $\langle i,n]\ t_{n+1}\langle h,n]^{-1}$ have a unique reduced expression. Moreover, an element of affine length at least $2$ has a unique reduced expression if and only if it is of the first type. Inserting the elements of affine length $1$ in the first type and second type sets would not have given us a partition of the set of those elements, as we can see in the example of $W^{c}(\tilde{B}_{4})$ listed in Appendix B. This is the reason why we handle them separately.

Remark 4.8**.**

We did not include $W^{c}(\tilde{B}_{3})$ in the theorem of classification because it is clear from the pushing algorithm in the Theorem that $W(\tilde{B}_{n})$ is an "affinization" of $W(D_{n})$ for $n\geq 4$, which is not the case of $W(\tilde{B}_{3})$ which can be looked at as an affinisation of $W(B_{2})$ or $W(A_{3})$. The classification of elements in $W^{c}(\tilde{B}_{3})$ is fairly easy, we leave it to the reader by recommending to consider our algorithm of pushing while sseing it as an affinisation of $W(B_{2})$. We choose to list in the Appendix $W(\tilde{B}_{4})$.*

4.3. Reduced expressions of elements of $W^{c}(\tilde{D}_{n+1})$

We call $\psi_{1}$ (resp. $\psi_{n}$) the automorphism of $W(\tilde{D}_{n+1})$ that exchanges $\sigma_{\bar{1}}$ with $\sigma_{1}$ (resp. $\sigma_{\bar{n}}$ with $\sigma_{n}$) and fixes the other generators.

Lemma 4.9**.**

Let $w$ be a fully commutative element in $W(\tilde{D}_{n+1})$ with $L(w)=m\geq 2$. Fix a reduced expression of $w$ as follows:

[TABLE]

with $u_{i}$, for $1\leq i\leq m+1$, a reduced expression of a fully commutative element in $W^{c}(D_{n+1})$. Then $u_{2},\dots,u_{m}$ are $\tilde{D}$-extremal elements, and there is a reduced expression of $w$ of the form:

[TABLE]

where $v_{m+1}\in W^{c}(D_{n+1})$, $i_{1}=n+1$ or $|i_{1}|\leq n$, $j_{1}=n$ or $|j_{1}|\leq n-1$, $|j_{1}|<i_{1}$ or $j_{1}=-i_{1}=\pm 1$, and for $2\leq t\leq m$ the elements $\langle i_{t},n]\langle j_{t},n-1]$ belong to the following list:

[TABLE]

Proof.

Since $\sigma_{\bar{n}}$ commutes with all the generators but $\sigma_{n-1}$, every two consecutive occurrences of $\sigma_{\bar{n}}$ must have in between an occurrence of $\sigma_{n-1}$, for the sake of keeping the expression reduced, and this occurrence must be properly more than once, for the sake of full commutativity; this shows that the $u_{i}$ with $2\leq i\leq m$ must be $\tilde{D}$-extremal.

Now we proceed as in the proof of Lemma 4.6: we use form (9) for $u_{1}$ and push the terms $\langle m_{i},n_{i}]$ with $n_{i}<n-1$ to the right of the leftmost $\sigma_{\bar{n}}$ (as well as the term $\sigma_{n}$ if $n_{1}=n$ and $n_{2}<n-1$, so that the term $\langle i_{1},n]\langle j_{1},n-1]$, if non trivial, has $|j_{1}|\leq n-1$), then we repeat this procedure with the new $u_{2}$ thus obtained and the $\sigma_{\bar{n}}$ appended at its right, and so on, until we get formally form (14). For the rest it is sufficient to see Definition 4.4 and notice that, in the $\tilde{D}$-extremal element $\langle-(n-1),n]=\langle-(n-1),n-1]\sigma_{n}$, the rightmost term $\sigma_{n}$ can also be pushed to the right of the $\sigma_{\bar{n}}$ that follows it on its right, so that this element is not needed in the list. ∎

Lemma 4.10**.**

Let $w$ be a fully commutative element in $W(\tilde{D}_{n+1})$ with $L(w)\!=m\!\geq 2$. We write $w$ in form (14) from Lemma 4.9. For $2\leq t\leq m$ we have either $i_{t}>0$ or $i_{t}=-1$. Furthermore:

1. (1)

If $j_{t}>1$ and $t\leq m-1$, then $|i_{t+1}|<j_{t}<i_{t}$. 2. (2)

If $-(n-1)<j_{t}<-1$ or $j_{t}=0$, then $t=m$ and $v_{m+1}=1$. 3. (3)

If $j_{t}=\pm 1$ then $i_{s}=-j_{t}=-j_{s}$ for all $s>t$, $s\leq m$. 4. (4)

If $j_{t}=-(n-1)$ then $j_{s}=-(n-1)$ for $2\leq s\leq m$ and $i_{s}=n$ for $2<s\leq m$ while $i_{2}=n$ or $n+1$.

Proof.

We refer to the element $v_{t}=\langle i_{t},n]\langle j_{t},n-1]$ according to list (15) in Lemma 4.9. We remark from that list that either $i_{t}>0$ or $i_{t}=-1$. We call an expression $\sigma_{a_{1}}\dots\sigma_{a_{u}}$ admissible if it is the reduced expression of a fully commutative element.

In case (1) the element $v_{t}$ has form (c) with $j>1$. Now $\langle i,n]\langle j,n-1]\sigma_{\bar{n}}\sigma_{s}$ is admissible if and only if $s<j$ or $\sigma_{s}=\sigma_{\bar{1}}$, since as before: $\sigma_{n}$ (resp. $\sigma_{n-1}$, resp. $\sigma_{u}$ with $j\leq u<n-1$) would produce the braid $\sigma_{n}\sigma_{n-1}\sigma_{n}$ (resp. $\sigma_{n-1}\sigma_{\bar{n}}\sigma_{n-1}$, resp. $\sigma_{u}\sigma_{u+1}\sigma_{u}$).

In case (2) the element $v_{t}$ has form (c) with $-(n-1)<j<-1$ or $j=0$. There is no generator $\sigma_{s}$ that can make $\langle i,n]\langle j,n-1]\sigma_{\bar{n}}\sigma_{s}$ admissible.

In case (3) indeed the element $v_{t}$ has form (c) or (d). Again $\sigma_{n}\langle-1,n-1]\sigma_{\bar{n}}\sigma_{s}$ is admissible if and only if $\sigma_{s}=\sigma_{1}$. Applying $\psi_{1}$ gives the other case.

In case (4) the element $v_{t}$ has form (a) or (b). We note that $\langle-(n-1),n-1]\sigma_{\bar{n}}\sigma_{s}$ is admissible if and only if $\sigma_{s}=\sigma_{n}$, so the only choice for $v_{t+1}$ is form (b). We now look for possible forms of $v_{t-1}$ using the previous cases : forms (c) and (d) cannot be followed by form (a) or (b); forms (a) and (b) are followed by (b). Hence $v_{2}$ may have form (a) while $v_{s}$ for $2<s\leq m$ must have form (b). ∎

These lemmas are the key to the normal form of fully commutative elements in $W(\tilde{D}_{n+1})$.

Theorem 4.11**.**

Let $w\in W^{c}(\tilde{D}_{n+1})$ with $L(w)\geq 2$. Then $w$ can be written in a unique way as a reduced word of one and only one of the following two forms, for non negative integers $p$ and $k$:

**First type: **

[TABLE]

*with $-n\leq f\leq n+1$, $-(n-1)\leq i\leq n$, $\epsilon=0$ or $1$, $\eta=0$ or $1$, $\epsilon\eta=0$ and if $\epsilon+\eta>0$ then $i=-(n-1)$. In other words the term $(\sigma_{n}\langle-(n-1),n-1]\;\sigma_{\bar{n}})^{k}$ may be preceded either by $\sigma_{n}\langle-(n-1),n-1]\sigma_{\bar{n}}$ or $\langle i,n-1]\sigma_{\bar{n}}$ with $-(n-1)\leq i\leq n$, or by $\sigma_{\bar{n}}\langle-(n-1),n-1]\sigma_{\bar{n}}$. *

**Second type: **

[TABLE]

with $w_{r}\in W^{c}(D_{n+1})$ and

• **•: **

if $p>0$: \ n+1\geq i_{1}>j_{1}>i_{2}>j_{2}...>i_{p}>|j_{p}|\geq 0,i_{p}>1,j_{p}\neq-1\;

• **•: **

if $p>0$ and $j_{p}\leq 0$ : $k=0$, $w_{r}=1$ and $j_{p}\neq-(n-1)$;

• **•: **

if $k=0$ and $j_{p}>1$: $w_{r}$ is of form (9) such that $|m_{1}|<j_{p}$;

• **•: **

*if $k>0$ or if $k=0$ and $j_{p}=1$: $w_{r}=\langle-1,r_{1}]\langle 1,r_{2}]\dots\langle(-1)^{u},r_{u}]$ with $1\leq r_{u}<\dots<r_{1}\leq n$ and $1,-1$ alternate; *

*or $w$ is the image under $\psi_{1}$ of such an element (in this case one must replace the condition $j_{p}\neq-1$ by $j_{p}\neq 1$). *

*The affine length of $w$ of the first (resp. second) type is $k+1+\epsilon$ (resp. $p+k$) and we have $0\leq p\leq\frac{n-1}{2}$.

Now suppose that $L(w)=1$, then it has a unique reduced expression of the form:

[TABLE]

• •

either $i=n+1$, $j=n$ and $v$ is arbitrary,

• •

or $|j|\leq n-1$, $i=-1$ or $1\leq i\leq n+1$, $|j|<i$ or $j=-i=\pm 1$ , and:

• –

if $|j|=1$ and $i\leq n$ then $\mathscr{L}(v)\subseteq\{\sigma_{-j}\}$ (with the convention $\sigma_{-1}=\sigma_{\bar{1}}$);

• –

if $|j|=1$ and $i=n+1$ then $\mathscr{L}(v)\subseteq\{\sigma_{-j},\sigma_{n}\}$;

• –

if $j<-1$ or $j=0$, and $i\leq n$, then $v=1$;

• –

if $j<-1$ or $j=0$, and $i=n+1$, then $v=\langle f,n]^{-1}$ with $-n\leq f\leq n+1$;

• –

if $j>1$ and $i\leq n$ then $|m_{1}|<j$;

• –

if $j>1$ and $i=n+1$ then either $|m_{1}|<j$ or, for some $s\leq r$, $m_{1}=n_{1}=n$, …, $m_{s}=n_{s}=n-s+1$, $|m_{s+1}|<j$.

Conversely, every $w$ of the above form is in $W^{c}(\tilde{D}_{n+1})$.

Proof.

Let $w$ be a fully commutative element in $W(\tilde{D}_{n+1})$ with $L(w)=m\geq 2$, written in form (14) from Lemma 4.9. We consider the terms $\langle i_{t},n]\langle j_{t},n-1]$ in $w$ for $2\leq t\leq m$.

If one of them has form (a) or (b), we know from Lemma 4.10 (4) that all of them have form (b) except possibly the first one that can have form (a), we thus get some power of $\sigma_{n}\langle-(n-1),n-1]\;\sigma_{\bar{n}}$: this is the first type. Same arguments as in the proof of this lemma show that the rightmost term $v_{m+1}$ has to be some right truncation of $\sigma_{n}\langle-(n-1),n-1]\sigma_{n}$, more easily written in the form $\langle f,n]^{-1}$ with $-n\leq f\leq n+1$. We now check the conditions on the introductory term $v_{1}=\langle i_{1},n]\langle j_{1},n-1]$.

If the term with $t=2$ has form (b) we have to find conditions for the expression $v_{1}\sigma_{\bar{n}}\sigma_{n}\langle-(n-1),n-1]$ to be admissible. If non trivial, $v_{1}$ ends with $\sigma_{n-1}$ by the assumption $|j_{1}|<i_{1}$ and if it contains a $\sigma_{n}$ (i.e. $|i_{1}|\leq n$) we must have $j_{1}=-(n-1)$ to forbid the braid $\sigma_{n}\sigma_{n-1}\sigma_{n}$ (remember that $\sigma_{\bar{n}}$ and $\sigma_{n}$ commute). So there is no choice but $i_{1}=n$. We thus have a left truncation of $\sigma_{n}\langle-(n-1),n-1]$.

If the term with $t=2$ has form (a) we inspect $v_{1}\sigma_{\bar{n}}\langle-(n-1),n-1]$. If non trivial, $v_{1}$ must end with $\sigma_{n}$, which is against our convention in Lemma 4.9, so $v_{1}$ is trivial, hence the form for the first type.

Otherwise those terms have form (c) or (d) and we know from Lemma 4.10 (1) that at most the $[n/2]$ first terms may have form (c), so eventually, for $t$ big enough, all the terms will have form (d), i.e. we get some power of $\langle-1,n]\langle 1,n-1]\;\sigma_{\bar{n}}$ or of its image under $\psi_{1}$: this is the second type. We check first the form of $v_{m+1}$. The proof of Lemma 4.10 (3), resp. (2), resp. (1), gives the case where $k>0$, resp. $k=0$ and $j_{p}\leq 0$, resp. $k=0$ and $j_{p}>0$. The form of the introductory term $\langle i_{1},n]\langle j_{1},n-1]\;\sigma_{\bar{n}}\dots\langle i_{p},n]\langle j_{p},n-1]\;\sigma_{\bar{n}}$ is an immediate consequence of the previous lemmas.

We have $n+1\geq i_{1}\geq i_{p}+2p\geq 2+2p$ hence $p\leq\frac{n-1}{2}$.

If $w$ has affine length $1$, as in the proof of Lemma 4.9, we use the fact that $\sigma_{\bar{n}}$ commutes with all generators except $\sigma_{n-1}$ to obtain the expression $\langle i,n]\langle j,n-1]\sigma_{\bar{n}}v$ where either $|j|\leq n-1$ or $j=n$ and $i=n+1$. The various cases are obtained in the same way as previously, we do not detail this. ∎

We note that the element of affine length $2$

[TABLE]

is a first type element. Inserting the elements of affine length $1$ in the first type and second type sets would not have given us a partition of the set of those elements as we can see in the example in $W^{c}(\tilde{D}_{4})$ given in Appendix C. This is the reason why we handle them separately.

5. The towers of fully commutative elements

When a Coxeter group is a parabolic subgroup of another, full commutativity is preserved by the natural injection. This is not the case for the embeddings defined in Corollary 2.2: the monomorphism $G_{n}:W(\tilde{D}_{n})\longrightarrow W(\tilde{D}_{n+1})$ does not preserve full commutativity, even for the simple reflexion $\sigma_{\bar{n-1}}$, while $L_{n}:W(\tilde{B}_{n})\longrightarrow W(\tilde{B}_{n+1})$ preserves the full commutativity of first type elements and elements of affine length 1 in $W^{c}(\tilde{B}_{n})$, but does not preserve it for $t_{n}\sigma_{n-1}t_{n}$, for example, in the set of second type fully commutative elements. We will use the normal form for fully commutative elements established in Theorem 4.11 (resp. Theorem 4.7) to produce embeddings from $W^{c}(\tilde{B}_{n})$ into $W^{c}(\tilde{B}_{n+1})$ and from $W^{c}(\tilde{D}_{n})$ into $W^{c}(\tilde{D}_{n+1})$.

5.1. The tower of $\tilde{B}$-type fully commutative elements

For $n>2$, we denote by $W^{c}_{1}(\tilde{B}_{n})$ the set of first type fully commutative elements in addition to fully commutative elements of affine length 1, and by $W^{c}_{2}(\tilde{B}_{n})$ the set of second type fully commutative elements. We thus have the following partition (under the convention $D_{3}:=A_{3}$):

[TABLE]

Definition 5.1**.**

For any $w\in W^{c}(\tilde{B}_{n})$ we define elements $I(w)$ and $J(w)$ of $W(\tilde{B}_{n+1})$ by the following expressions:

• •

if $w\in W^{c}_{2}(\tilde{B}_{n})$, then $I(w)$ (resp. $J(w)$) is obtained by substituting $\sigma_{n}t_{n+1}$ (resp. $t_{n+1}\sigma_{n}$) to $t_{n}$ in the normal form (12) for $w$;

• •

if $w\in W^{c}_{1}(\tilde{B}_{n})$, then $I(w)=J(w)$ is obtained by substituting $\sigma_{n}t_{n+1}\sigma_{n}$ to $t_{n}$ in the normal form (11) or (13) for $w$;

• •

if $w\in W^{c}(D_{n})$, then $I(w)=J(w)=w$.

Theorem 5.2**.**

For any $w\in W^{c}(\tilde{B}_{n})$, the expressions for $I(w)$ and $J(w)$ in Definition 5.1 are reduced and they are reduced expressions for fully commutative elements in $W(\tilde{B}_{n+1})$. The maps thus defined:

[TABLE]

are injective, preserve the affine length and satisfy

[TABLE]

*The injections $I$ and $J$ map first type (resp. second type) elements to first type (resp. second type) elements and their images intersect exactly on $I(W^{c}_{1}(\tilde{B}_{n})\bigsqcup W^{c}(D_{n}))$. *

Proof.

The proof is mutatis mutandis the proof of [4, Theorem 5.2]. We will only emphasize the main fact, which must be kept in mind for the applications. For an element $w$ of the second type, the expression of $J(w)$ is obtained by substituting $t_{n+1}\sigma_{n}$ to $t_{n}$ in the normal form for $w$. In order to find the normal form of $J(w)$ one uses the property that $t_{n+1}$ commutes with $\sigma_{\bar{1}}$ and $\sigma_{i}$ for $1\leq i<n$, hence each $t_{n+1}$ can be pushed to its left until either it is stopped by a $\sigma_{n}$ or it is in the leftmost position. In particular, the leftmost term in the normal form of $J(w)$ is $t_{n+1}$ whereas no reduced expression of $I(w)$ can have $t_{n+1}$ as its leftmost term (since the expression $t_{n+1}I(w)$ is also a normal form of the second type). Consequently the images $I(W^{c}_{2}(\tilde{B}_{n}))$ and $J(W^{c}_{2}(\tilde{B}_{n}))$ are disjoint. ∎

We remark that the injections $I$ and $J$ on $W^{c}_{1}(\tilde{C}_{n})\bigsqcup W^{c}(B_{n})$ are but the restriction of $L_{n}$ in (2.2). Actually $I$ and $J$ may be defined on all $W(\tilde{B}_{n})$, but as we don’t need this, we won’t examine it further.

5.2. The tower of $\tilde{D}$-type fully commutative elements

For $n>3$, we denote by $W^{c}_{1}(\tilde{D}_{n})$ the set of first type fully commutative elements in addition to fully commutative elements of affine length 1, and by $W^{c}_{2}(\tilde{D}_{n})$ the set of second type fully commutative elements. We thus have the following partition:

[TABLE]

Definition 5.3**.**

For any $w\in W^{c}(\tilde{D}_{n})$ we define elements $I(w)$ and $J(w)$ of $W(\tilde{D}_{n+1})$ by the following expressions:

• •

if $w\in W^{c}_{2}(\tilde{D}_{n})$, then $I(w)$ (resp. $J(w)$) is obtained by substituting $\sigma_{n}\sigma_{n-1}\sigma_{\bar{n}}$ (resp. $\sigma_{\bar{n}}\sigma_{n-1}\sigma_{n}$) to $\sigma_{\bar{n-1}}$ in the normal form (17) for $w$;

• •

if $w\in W^{c}_{1}(\tilde{D}_{n})$, then $I(w)=J(w)$ is obtained by substituting $\sigma_{n-1}\sigma_{\bar{n}}\sigma_{n}$ to $\sigma_{\bar{n-1}}$ in the normal form (16) or (18) for $w$, with the following exceptions:

• –

if $w$ is a first type element with $\epsilon=1$ then the leftmost term $\sigma_{\bar{n-1}}$ in the normal form (16) must be substituted by $\sigma_{n}\sigma_{\bar{n}}\sigma_{n-1}$;

• –

if $w$ has affine length $1$ with $i\neq n$, then $\sigma_{\bar{n-1}}$ in (18) must be substituted by $\sigma_{n}\sigma_{n-1}\sigma_{\bar{n}}$.

• •

if $w\in W^{c}(D_{n})$, then $I(w)=J(w)=w$.

Theorem 5.4**.**

For any $w\in W^{c}(\tilde{D}_{n})$, the expressions for $I(w)$ and $J(w)$ in Definition 5.3 are reduced and they are reduced expressions for fully commutative elements in $W(\tilde{D}_{n+1})$. The maps thus defined:

[TABLE]

are injective, preserve the affine length and satisfy

[TABLE]

*The injections $I$ and $J$ map first type (resp. second type) elements to first type (resp. second type) elements and their images intersect exactly on $I(W^{c}_{1}(\tilde{D}_{n})\bigsqcup W^{c}(D_{n}))$.

Proof.

We start with $w\in W^{c}_{2}(\tilde{D}_{n})$ given in form (17) and notice that, since $\sigma_{n}$ commutes with all generators except $\sigma_{n-1}$, we have :

[TABLE]

so that $I(w)$ is indeed the reduced expression of an element in $W^{c}_{2}(\tilde{D}_{n+1})$ with the same parameters $i_{1},\dots,i_{p},j_{1},\dots,j_{p},k,w_{r}$, as $w$. On the other hand, again using commutation relations, we find:

[TABLE]

Hence $J(w)$ is also the reduced expression of an element in $W^{c}_{2}(\tilde{D}_{n+1})$, here with a shift in parameters: if we denote with primes the parameters for $J(w)$ we have

[TABLE]

with $x=j_{p}$ if $k=0$, or $x=\pm 1$, so $x\neq n+1$. In particular this shift acts on the sequence of $1$ and $-1$, so $J(w)$ will be the image under $\psi_{1}$ of an element in form (17).

The injectivity of both maps is clear from the unicity of the normal form, since the parameters of the images determine the parameters of the source element. Furthermore we remark that in the normal form for $I(w)$, the term on the right of the rightmost $\sigma_{\bar{n}}$ is $w_{r}$ which belongs to $W^{c}(D_{n})$ whereas for $J(w)$ the corresponding term is $\langle x,n]w_{r}$ which contains $\sigma_{n}$, so the images of $I$ and $J$ on second type elements do not intersect.

We now turn to first type elements and take $w$ given in form (16). Here

[TABLE]

so, if $\epsilon=0$, $I(w)$ is the reduced expression of a first type element, with the same parameters $\epsilon,\eta,i,k,f$ if $\eta=0$, while if $\eta=1$ (hence $i=-(n-2)$) the parameters of $I(w)$, written with a prime, are $\epsilon^{\prime}=\epsilon,\eta^{\prime}=0,i^{\prime}=-(n-1),k^{\prime}=k,f^{\prime}=f$. If $\epsilon=1$, our element starts with $\sigma_{\bar{n-1}}\langle-(n-2),n-2]\sigma_{\bar{n-1}}$ and the previous substitution for the leftmost $\sigma_{\bar{n-1}}$ would produce, again for commutation reasons, the braid $\sigma_{n}\sigma_{n-1}\sigma_{n}$. We substitute instead $\sigma_{\bar{n}}\sigma_{n}\sigma_{n-1}$, getting:

[TABLE]

hence a first type element with parameters $\epsilon^{\prime}=\eta^{\prime}=0$, $i^{\prime}=n$, $k^{\prime}=k+1$ and $f^{\prime}=f$. The set of such parameters is disjoint from the set obtained with $\epsilon=0$, indeed for those we had $i^{\prime}\leq n-1$, injectivity for first type follows.

The last case to consider is elements of affine length $1$. Here substituting $\sigma_{n-1}\sigma_{\bar{n}}\sigma_{n}$ to $\sigma_{\bar{n-1}}$ in the normal form (16) transforms $\langle i,n-1]\langle j,n-2]\sigma_{\bar{n-1}}\ v$ into $\langle i,n-1]\langle j,n-1]\sigma_{\bar{n}}\sigma_{n}\ v$ which is indeed, if $i=n$, the reduced expression of a fully commutative element in the required form (16), with parameters $i^{\prime}=n+1$, $j^{\prime}=j$ and $v^{\prime}=\sigma_{n}v$. If $i<n$ the correct substitution is $\sigma_{n}\sigma_{n-1}\sigma_{\bar{n}}$, resulting in $\langle i,n]\langle j,n-1]\sigma_{\bar{n}}\ v$ that again has the required form, with parameters $i^{\prime}=i$, $j^{\prime}=j$ and