Algebraic Definition for Coxeter Group

Coxeter Matrix

Let $S$ be a set. A matrix $m:S\times S\to \{1,2,\dots ,\infty \}$ is called a Coxeter matrix if it satisfies $m(s,s^{\prime })=m(s^{\prime },s);$ $m(s,s^{\prime })=1⟺s=s^{\prime }.$

Equivalently, $m$ can be represented by a Coxeter graph (or Coxeter diagram) whose node set is $S$ and whose edges are the unordered pairs $\{s,s^{\prime }\}$ such that $m(s,s^{\prime })\ge 3$. The edges with $m(s,s^{\prime })\ge 4$ are labeled by that number. Let $S_{\text{fin}}^2=\{(s,s^{\prime })\in S^2:m(s,s^{\prime })\ne \infty \}$. A Coxeter matrix $m$ determines a group $W$ with the presentation

\begin{cases} \text{Generators: } S \\ \text{Relations: } (ss')^{m(s,s')} = e, \quad \text{for all } (s, s') \in S_{\text{fin}}^2. \end{cases} \tag{1.3}

Definition

We call $(W,S)$ defined as above a Coxeter System.

Proposition

The following three statements are equivalent and make explicit what it means for $W$ to be determined by $m$ via the presentation :

1. (Universality Property) If $G$ is a group and $f:S\to G$ is a mapping such that $(f(s)f(s^{\prime }){)}^{m(s,s^{\prime })}=e$ for all $(s,s^{\prime })\in S_{\text{fin}}^2$, then there is a unique extension of $f$ to a group homomorphism $\bar{f}:W\to G$.
2. $W\cong F/N$, where $F$ is the free group generated by $S$ and $N$ is the normal subgroup generated by $\{(s{s}^{\prime }{)}^{m(s,s^{\prime })}:(s,s^{\prime })\in S_{\text{fin}}^2\}$.
3. Let $S^{\ast }$ be the free monoid generated by $S$ (i.e., the set of words in the alphabet $S$ with concatenation as product). Let $\equiv$ be the equivalence relation generated by allowing insertion or deletion of any word of the form $(s{s}^{\prime }{)}^{m(s,s^{\prime })}=\underset{2m(s,s^{\prime })}{\underbrace{s{s}^{\prime }s{s}^{\prime }s\dots s^{\prime }s{s}^{\prime }}}$ for $(s,s^{\prime })\in S_{\text{fin}}^2$. Then, $S^{\ast }/\equiv$ forms a group isomorphic to $W$.

Proposition

Let $(W,S)$ be the Coxeter system determined by a Coxeter matrix $m$. Let $s$ and $s^{\prime }$ be distinct elements of $S$. Then, the following hold:

• (i) (The classes of) $s$ and $s^{\prime }$ are distinct in $W$.
• (ii) The order of $s{s}^{\prime }$ in $W$ is $m(s,s^{\prime })$.

Theorem

Up to isomorphism there is a one-to-one correspondence between Coxeter matrices and Coxeter systems.

Definition

$T:=ws{w}^{-1}:s\in S,w\in W$ $t_i\stackrel{\text{def}}{=}{s}_1{s}_2\dots s_{i-1}{s}_i{s}_{i-1}\dots s_2{s}_1,\text{for }1\le i\le k,$ $\hat{T}(s_1{s}_2\dots s_k)\stackrel{\text{def}}{=}(t_1,t_2,\dots ,t_k).$ Left Reflection Set: $T_L(w)\stackrel{\text{def}}{=}\{t\in T:\ell (tw)<\ell (w)\}$ Right Reflection Set: $T_R(w)\stackrel{\text{def}}{=}\{t\in T:\ell (wt)<\ell (w)\}$ Left Descent Set: $D_L(w)\stackrel{\text{def}}{=}{T}_L(w)\cap S$ Right Descent Set: $D_R(w)\stackrel{\text{def}}{=}{T}_R(w)\cap S$

Lemma

If $w=s_1{s}_2\dots s_k$, with $k$ minimal, then $t_i\ne t_j$ for all $1\le i<j\le k$.

Theorem

(i) The mapping $s\mapsto {\pi }_s$ extends uniquely to an injective homomorphism $w\mapsto {\pi }_w$ from $W$ to $S(R)$. (ii) ${\pi }_t(t,\epsilon )=(t,-\epsilon )$, for all $t\in T$.

Proposition

This proposition describes how the length of an element $w\in W$ (the minimal number of simple reflections $s\in S$ whose product is $w$) interacts with group multiplication.

• (i) $\epsilon (w)=(-1{)}^{\ell (w)}$: The sign function $\epsilon :W\to \{1,-1\}$ is a homomorphism, where $\ell (w)$ parity determines the sign.
• (ii) $\ell (uw)\equiv \ell (u)+\ell (w)(\mathrm{mod}2)$: Generalizes the parity rule to products of arbitrary elements.
• (iii) $\ell (sw)=\ell (w)\pm 1$: Multiplying by a simple reflection $s$ changes the length by exactly 1.
• (iv) $\ell (w^{-1})=\ell (w)$: Inverting an element preserves its length.
• (v) Triangle Inequality: $|\ell (u)-\ell (w)|\le \ell (uw)\le \ell (u)+\ell (w)$.
• (vi) Word Metric: The function $d(u,w)=\ell (u^{-1}w)$ defines a metric on the Cayley graph of the Coxeter system $(W,S)$.

Strong Exchange Property

Suppose $w=s_1{s}_2\dots s_k$ ($s_i\in S$) and $t\in T$. If $\ell (tw)<\ell (w)$, then $tw=s_1\dots {\hat{s}}_i\dots s_k$ for some $i\in [k]$.

Corollary

If $w=s_1{s}_2\dots s_k$ is reduced and $t\in T$, then the following are equivalent: (a) $\ell (tw)<\ell (w)$, (b) $tw=s_1\dots {\hat{s}}_i\dots s_k$, for some $i\in [k]$, (c) $t=s_1{s}_2\dots s_i\dots s_2{s}_1$, for some $i\in [k]$.

Corollary

$|T_L(w)|=\ell (w)$.

Corollary

For all $s\in S$ and $w\in W$, the following hold: (i) $s\in D_L(w)$ if and only if some reduced expression for $w$ begins with the letter $s$. (ii) $s\in D_R(w)$ if and only if some reduced expression for $w$ ends with the letter $s$.

Deletion Property

If $w=s_1{s}_2\dots s_k$ and $\ell (w)<k$, then $w=s_1\dots {\hat{s}}_i\dots {\hat{s}}_j\dots s_k$ for some $1\le i<j\le k$.

Corollary

(i) Any expression $w=s_1{s}_2\dots s_k$ contains a reduced expression for $w$ as a subword, obtainable by deleting an even number of letters. (ii) Suppose $w=s_1{s}_2\dots s_k=s_1^{\prime }{s}_2^{\prime }\dots s_k^{\prime }$ are two reduced expressions. Then, the set of letters appearing in the word $s_1{s}_2\dots s_k$ equals the set of letters appearing in $s_1^{\prime }{s}_2^{\prime }\dots s_k^{\prime }$. (iii) $S$ is a minimal generating set for $W$; that is, no Coxeter generator can be expressed in terms of the others.