Coxeter Systems

Abstract

This chapter aims essentially to fix notation and to gather, without proofs (and maybe sometimes in a way that is more convenient for our needs but which might not reflect the logical development of the theory), some of the results about Coxeter groups that will be needed throughout this book.

Appendices

Notes

As the reader might have noticed, we have mainly followed the treatment of [Bou] or [GePf, Chapters 1 and 2]. An exception is Section 3.10, where fixed points subgroups under Coxeter system automorphisms are investigated: there are several proofs of Theorem 3.10.6 (see [Ste, §11] for W finite and [Hée, §3], [Lus20, Appendix] or [GeIa2, Theorem 1] for the general case). Our argument, which is based on the study of reduced expressions, mostly follows [GeIa2], but is slightly different.

Fig. 3.e.

Type ${\mathbf H}{\mathbf y}{\mathbf p}(4,4,4)$

Exercises

Exercises (or questions) marked with a star are more difficult than the others.

Exercise 3.1

(Type ${\varvec{G_2}}$). Assume here that $S=\{s, t\}$ and $m_{st}=6$, so that (W, S) is of type $G_2=I_2(6)$. Here is the corresponding Coxeter graph:

Show that s and t are not conjugate in W, that

$$W=\{1,s,t,st,ts,sts,tst,stst,tsts,ststs,tstst, ststst=tststs=w_0\}$$

and that $C_W(s) = \langle s, tstst \rangle \simeq {\mathbb {Z}}/2{\mathbb {Z}}\times {\mathbb {Z}}/2{\mathbb {Z}}$. Check that $C_W(s) \ne W_{s^\perp }$ (compare with Proposition 3.8.2).

Show that W is isomorphic to the dihedral group of order 12. Compute $X_s$ and $X_t$; show that ${\mathrm {Z}}(W)=\{w_0\}$.

Exercise 3.2

(Direct products). Let $S=S_1 \dot{\cup }S_2$ be a partition such that, if $s_1 \in S_1$ and $s_2 \in S_2$, then $m_{s_1s_2}=2$ (i.e. $s_1s_2=s_2s_1$). Let $W_i=W_{S_i}$ ($i \in \{1,2\}$).

• (a) Show that $W=W_1 \times W_2$.

• (b) Let $E_i$ be the reflection representation associated with the Coxeter system $(W_i, S_i)$. Show that $E \simeq E_1 \oplus E_2$ and that $E_i$ is W-stable.

• (c) Let $(w_1,w_2) \in W_1 \times W_2$. Show that $\ell (w_1w_2)=\ell (w_1)+\ell (w_2)$.

• (d) Show that $X_{S_2}=W_1$ and $X_{S_1}=W_2$.

• (e) Let $T_i$ be the set of reflections of $W_i$. Show that $T=T_1 \dot{\cup }T_2$.

• (f) Let $(x_1,x_2)$ and $(y_1,y_2)$ be two elements of $W_1 \times W_2$. Show that $x_1x_2 \mathop {{\varvec{\leqslant }}}\nolimits y_1y_2$ if and only if $x_1 \mathop {{\varvec{\leqslant }}}\nolimits y_1$ and $x_2 \mathop {{\varvec{\leqslant }}}\nolimits y_2$.

Exercise 3.3.

Show that the map $N : W \rightarrow {\mathscr {P}}(T)$ is injective. (Hint: Use Corollary 3.3.5 and an induction argument based on the length function.)

Exercise 3.4

(Abelianization). Let ${\bar{S}}$ denote the quotient of S by the equivalence relation linking two elements s and t of S if they are conjugate in W. If I is a W-closed subset of S, we identify it with its image ${\bar{I}}$ in ${\bar{S}}$ and, if ${\bar{I}}=\{{\bar{s}}_1,{\bar{s}}_2,\dots ,{\bar{s}}_r\}$ with $r =|{\bar{I}}|$, we denote by $d_I$ the image of $s_1s_2\dots s_r$ in $W/{\mathrm {D}}(W)$, where we recall that ${\mathrm {D}}(W)$ is the derived subgroup of W.

• (a) Show that $d_I$ depends only on I and not on the choice of the representatives $s_1$,..., $s_r$.

• (b) We identify ${\mathscr {P}}({\bar{S}})$ with the set of W-closed subsets of S as before. Show that the map

  $$\begin{array}{rccc} {d} : &{} {({\mathscr {P}}({\bar{S}}),\dotplus )} &{} \longrightarrow &{} {W/{\mathrm {D}}(W)} \\ &{} {I} &{} \longmapsto &{} {d_I} \end{array}$$

  is an isomorphism of groups.

Exercise 3.5

(Linear characters). We keep the notation of Exercise 3.4. Let I be a W-closed subset of S. We define $\varepsilon _I(w)=(-1)^{\ell _I(w)}$. Note that $\varepsilon _\varnothing = {\varvec{1}}_W$ (where ${\varvec{1}}_W(w)=1$) and $\varepsilon _S=\varepsilon$.

• (a) Show that $\varepsilon _I : W \rightarrow \{1,-1\}$ is a morphism of groups.

• (b) Show that $\varepsilon _I\varepsilon _J = \varepsilon _{I \dotplus J}$.

• (c) We identify ${\mathscr {P}}({\bar{S}})$ with the set of W-closed subsets of S as in Exercise 3.4. Show that the map

  $$\begin{array}{ccc} {({\mathscr {P}}({\bar{S}}),\dotplus )} &{} \longrightarrow &{} {{{\mathrm{{\mathrm {Hom}}}}}(W,\{1,-1\})} \\ {I} &{} \longmapsto &{} {\varepsilon _I} \end{array}$$

  is an isomorphism of groups.

Exercise 3.6

(Bourbaki). Let $W^+ = {{\mathrm{{\mathrm {Ker}}}}}(\varepsilon )$. Fix $s \in S$ and, for $t \in S \setminus \{s\}$, set $g_t=ts$. Show that $W^+$ admits the following presentation:

$$ {\left\{ \begin{array}{ll} \mathrm{Generators\!\!:} &{} (g_t)_{t \in S\setminus \{s\}}\\ \mathrm{Relations\!\!:} &{} \forall t \in S\setminus \{s\},~g_t^{m_{ts}}=1\\ &{} \forall t \ne t' \in S \setminus \{s\},~(g_{t'}g_t^{-1})^{m_{tt'}}=1. \end{array}\right. } $$

Exercise 3.7.

Let s, $t \in S$ and $x \in W$ be such that $sx {\varvec{<}}x {\varvec{<}}tx {\varvec{<}}stx$. Show that ${\mathscr {R}}(x)={\mathscr {R}}(tx)$. (Hint: Use Deodhar’s Lemma 3.6.12 or Corollary 3.6.13.)

Exercise 3.8.

$^*$. If $\begin{pmatrix} a &{} b \\ c &{} d \end{pmatrix} \in {\mathbf G}{\mathbf L}_2({\mathbb {Z}})$, we denote by $\left[ \begin{array}{cccccccccc}a &{} b \\ c &{} d\end{array}\right]$ its image in ${\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {Z}})$. Note that the determinant $\det : {\mathbf G}{\mathbf L}_2({\mathbb {Z}}) \rightarrow \{1,-1\}$ factorizes through $\det _0: {\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {Z}}) \rightarrow \{1,-1\}$. For instance, ${\mathbf P}{\mathbf S}{\mathbf L}_2({\mathbb {Z}}) = {{\mathrm{{\mathrm {Ker}}}}}(\det _0)$. Now, let

$$s=\left[ \begin{array}{cccccccccc}1 &{} 0 \\ 0 &{} -1\end{array}\right] , \quad t=\left[ \begin{array}{cccccccccc}1 &{} 0 \\ -1 &{} -1\end{array}\right] \quad \mathrm{and}\quad u=\left[ \begin{array}{cccccccccc}0 &{} 1 \\ 1 &{} 0\end{array}\right] .$$

We denote by ${\mathrm {I}}_2$ the $2 \times 2$ identity matrix.

The aim of this exercise is to show that $({\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {Z}}),\{s,t, u\})$ is a Coxeter system.

• (a) Show that $s^2=t^2=u^2=1$ and that $(su)^2=(tu)^3=1$.

• (b) Show that st has infinite order.

• (c*) Show that ${\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {Z}})$ is generated by s, t and u.

Let (W, S) be the Coxeter system such that $S=\{{\mathbf s},{\mathbf t},{\mathbf u}\}$ and $m_{{\mathbf s}{\mathbf t}}=\infty$, $m_{{\mathbf s}{\mathbf u}}=2$ and $m_{{\mathbf t}{\mathbf u}}=3$. In other words, the Coxeter graph ${{\mathscr {C}}\!\!\,o\!\!\,x}(W, S)$ is given by

By (a) and (b), there exists a unique morphism of groups $p : W \longrightarrow {\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {Z}})$ such that $p({\mathbf s})=s$, $p({\mathbf t})=t$ and $p({\mathbf u})=u$. By (c), it is surjective. It remains to show that p is injective.

For this, let (x, y) be the canonical basis of ${\mathbb {R}}^2$, let ${\mathscr {E}}={\mathrm {S}}^2({\mathbb {R}}^2)$ be the second symmetric power (note that $(x^2,xy, y^2$) is an ${\mathbb {R}}$-basis of ${\mathscr {E}}$), let ${{\mathrm{{\mathrm {disc}}}}}: {\mathscr {E}}\rightarrow {\mathbb {R}}$ be the quadratic form defined by ${{\mathrm{{\mathrm {disc}}}}}(a x^2 + 2b xy + c y^2)=b^2-ac$ (the discriminant) and let $\delta : {\mathscr {E}}\times {\mathscr {E}}\rightarrow {\mathbb {R}}$ be the associated symmetric bilinear form. Let

$$e_s=2xy,\quad e_t = -2xy + y^2\quad \mathrm{and}\quad e_u = x^2-y^2.$$

Finally, we denote by ${\mathrm {S}}^2 : {\mathbf G}{\mathbf L}_2({\mathbb {R}}) \rightarrow {\mathbf G}{\mathbf L}_{\mathbb {R}}({\mathscr {E}})$ the canonical morphism of groups.

• (d) Show that ${{\mathrm{{\mathrm {disc}}}}}({\mathrm {S}}^2(\gamma ) f) = \det (\gamma )^2 {{\mathrm{{\mathrm {disc}}}}}(f)$ for all $\gamma \in {\mathbf G}{\mathbf L}_2({\mathbb {R}})$ and $f \in {\mathscr {E}}$.

• (e) Show that ${{\mathrm{{\mathrm {Ker}}}}}({\mathrm {S}}^2) = \{{\mathrm {I}}_2,-{\mathrm {I}}_2\}$. We denote by $\bar{{\mathrm {S}}} : {\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {Z}}) \rightarrow {\mathbf G}{\mathbf L}_{\mathbb {R}}({\mathscr {E}})$ the factorization of the restriction of ${\mathrm {S}}^2$ to ${\mathbf G}{\mathbf L}_2({\mathbb {Z}})$.

• (f) Show that the representation $\bar{{\mathrm {S}}}\,\circ \, p : W \rightarrow {\mathbf G}{\mathbf L}_{\mathbb {R}}({\mathscr {E}})$ is isomorphic to the reflection representation of W.

• (g) Deduce that p is injective and that $({\mathbf P}{\mathbf G}{\mathbf L}_2({\mathbb {Z}}),\{s,t, u\})$ is a Coxeter system.

• (h) Show that $\det \circ p = \varepsilon _W$.

• (i) Let $a=su$ and $b=tu$. Show that ${\mathbf P}{\mathbf S}{\mathbf L}_2({\mathbb {Z}})$ has the following presentation:

  $$ {\left\{ \begin{array}{ll} \mathrm{Generators\!\!:} &{} a, b \\ \mathrm{Relations\!\!:} &{} a^2=b^3=1. \end{array}\right. } $$

  (Hint: Use Exercise 3.6.)

Exercise 3.9

$^*\,$(Coxeter element). Assume here that S is finite, with cardinality n, and write $S=\{s_1,s_2,\dots , s_n\}$. Let $c=s_1s_2\cdots s_n \in W$ (c is called a standard Coxeter element). One of the aims of this exercise is to prove that c has finite order if and only if W is finite¹ (a result which is due to Howlett [Howl, Theorem 4.1]). We also study the eigenvalues of c in the reflection representation.

For simplicity, we set $e_i=e_{s_i} \in E$. Let $A=(a_{ij})_{1 \mathop {\leqslant }\nolimits i, j \mathop {\leqslant }\nolimits n}$ be the matrix defined by:

$$a_{ij}= {\left\{ \begin{array}{ll} 1 &{} \mathrm{if}\,\, i=j, \\ 0 &{} \mathrm{if}\,\, 1 \mathop {\leqslant }\nolimits j< i \mathop {\leqslant }\nolimits n,\\ 2B_W(e_i, e_j) &{} \mathrm{if}\,\, 1 \mathop {\leqslant }\nolimits i < j \mathop {\leqslant }\nolimits n.\\ \end{array}\right. } $$

Let B be the symmetric matrix $\bigl (2B_W(e_i, e_j)\bigr )_{1 \mathop {\leqslant }\nolimits i, j \mathop {\leqslant }\nolimits n}$. Note that $\det (A)=1$, that A is invertible and that

$$A + {A}^{t}A=B.$$

Finally, let C denote the matrix of c (for its action on E) in the basis ${\mathscr {E}}=(e_1,e_2,\dots , e_n)$ of E and let

$$\chi ({\mathbf t})=\det ({\mathbf t}{\mathrm {I}}_n - C)$$

denote its characteristic polynomial.

• (a) Show that $A^{-1}$ is the matrix of the linear map $\psi : E \rightarrow E$ such that $\psi (e_i)=s_1s_2\cdots s_{i-1}(e_i)$ for $1 \mathop {\leqslant }\nolimits i \mathop {\leqslant }\nolimits n$.

• (b) Show that $- {A}^{t}A$ is the matrix of the linear map $\psi ' : E \rightarrow E$ defined by $\psi '=c \psi ^{-1}$.

• (c) Deduce that $C=- {A}^{t}A ~A^{-1}$.

• (d) Show that ${C}^{t}C^{-1}=A^{-1} C A$ and that ${\mathbf t}^n\chi ({\mathbf t}^{-1})=\chi ({\mathbf t})$.

• (e) Show that $\chi ({\mathbf t}) = \det ({\mathbf t}A + {A}^{t}A)$.

• (f) Deduce that 1 is an eigenvalue of C if and only if $B_W$ is degenerate. If so, show that the multiplicity of 1 as an eigenvalue of C is then even (Hint: use (d)).

From now on, we set, for $\lambda \in {\mathbb {R}}_{\mathop {\geqslant }\nolimits 0}$,

$$M(\lambda )=(1+\lambda ) {\mathrm {I}}_n - \lambda A - {A}^{t}A.$$

We denote by $\rho (\lambda )$ the spectral radius of $M(\lambda )$ (i.e., if $\mu _1$, $\mu _2$,..., $\mu _r$ denote the complex eigenvalues of $M(\lambda )$, then $\rho (\lambda )=\max (|\mu _1|,|\mu _2|,\dots ,|\mu _r|)$). Recall that $\rho (\lambda )$ is a continuous function of $\lambda$.

• (g) Show that the coefficients of $M(\lambda )$ are non-negative.

• (h) Show that $\rho (0)=0$.

Note that, by the Perron–Frobenius Theorem, this implies that $\rho (\lambda )$ is an eigenvalue of $M(\lambda )$; moreover, if $M(\lambda )$ is irreducible,² then the multiplicity of $\rho (\lambda )$ as an eigenvalue of $M(\lambda )$ is equal to one.

• (h) Assume here that W is not finite (by Theorem 3.12.1, this implies that $B_W$ is not positive definite or, in other words, that $B=A + {A}^{t}A$ admits a non-positive eigenvalue).

  • (h1) Show that $\rho (1) \mathop {\geqslant }\nolimits 2$.

  • (h2) Deduce that there exists a $\lambda _0 \in {\mathbb {R}}$, $0 < \lambda _0 \mathop {\leqslant }\nolimits 1$, such that $\rho (\lambda _0)=1+\lambda _0$.

  • (h3) Deduce from (e) that $\lambda _0$ is an eigenvalue of C.

  • (h4) Prove that c has infinite order. (Hint: if $\lambda _0=1$, then, after reduction to the case where (W, S) is irreducible, use (f) and the fact that ${\rho (1)=2}$ is an eigenvalue of multiplicity one of M(1) to show that C is not diagonalizable over ${\mathbb {C}}$.)

From now on, and until the end of this exercise, we assume that ${{\mathscr {C}}\!\!\,o\!\!\,x}(W, S)$ is a forest. By Proposition 3.12.6, this implies that there exists a partition $\{1,2,\dots , n\}=I \dot{\cup }J$ such that $m_{s_i s_{i'}}=m_{s_js_{j'}}=2$ for all i, $i' \in I$ and j, $j' \in J$. We set $B'=B-2{\mathrm {I}}_n$ and

$$P({\mathbf t})=\det ({\mathbf t}{\mathrm {I}}_n + B').$$

For $\alpha \in {\mathbb {C}}$, let $d(\alpha )$ denote the diagonal matrix whose (i, i)-entry is equal to $\alpha$ if $i \in I$ and to 1 if $i \in J$.

• (i) Show that $d(\alpha ) (\alpha ^2 A + {A}^{t}A) d(\alpha )^{-1} = \alpha \bigl ((\alpha -1) {\mathrm {I}}_n + {A}^{t}A + A\bigr )$.

• ( j) Deduce that $B'$ is equivalent to $-B'$ and that, for $\alpha \ne 0$,

  $$\det (\alpha ^2 {\mathrm {I}}_n - C) = \alpha ^n P(\alpha +\alpha ^{-1}).$$

• (k) Deduce that, if $\beta$ is a complex eigenvalue of c which is not real, then $|\beta |=1$.