Representation Theory, Reflection groups and Groups of Lie Type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 8 October 2012

Representation Theory

An algebra is a vector space with a product so that $A$ is a ring.

Representation theory is the study of the category of $A$-modules (vector spaces $M$ with an action of $A\text{).}$

A simple $A$-module is an $A$-module $M$ with no submodules, except $D$ and $M\text{.}$

Problem: Determine the simple $A$-modules.

An indecomposable module is an $A$-module $M$ such that there DOES NOT EXIST $N$ and $P$ nonzero submodules with $M=N\oplus P\text{.}$

$$\begin{array}{cc} N P & N P \\ M=N\oplus P& \begin{array}{c}0⟶P⟶M⟶N⟶0\\ \text{but}M\ne N\oplus P\end{array}\end{array}$$

Reflection groups

Let $Z$ be a subring of $ℂ\text{.}$

A reflection group is a pair $({𝔥}_Z,W_0)$ with

1. ${𝔥}_Z$ a free $Z$-module
2. $W_0$ a finite subgroup of $GL\left({𝔥}_Z\right)$ generated by reflections.

A reflection is an element $s\in G{L}_n\left(ℂ\right)$ conjugate to

$$\left(\begin{array}{cccc}\xi & & & 0\\ & 1\\ & & \ddots \\ 0& & & 1\end{array}\right)\text{with}\xi \ne 1\text{.}$$

A crystallographic reflection group is a $\mathbb{Z}$-reflection group.

A Euclidean reflection group is an $ℝ$-reflection group.

Examples

Type $S{L}_3:$ ${𝔥}_{\mathbb{Z}}=\mathbb{Z}\text{-span}\{{\alpha }_1,{\alpha }_2\}$

$$W_0=⟨s_1{s}_2\mid s_i^2=1,s_1{s}_2{s}_1=s_2{s}_1{s}_2⟩$$ $$𝔥α1∨ 𝔥α2∨ 𝔥α3∨ s1 s2 α2 α1 s1s2 s2s1 s1s2s1=s2s1s2 C0$$

Type $G{L}_n:$ ${𝔥}_{\mathbb{Z}}=\mathbb{Z}{\epsilon }_1+\dots +\mathbb{Z}{\epsilon }_n\text{.}$

$$W_0=S_n\text{permuting}{\epsilon }_1,\dots ,{\epsilon }_n$$

Reflections in $S_n:$

$$\begin{array}{ccc}{s}_{ij}=& 1 ˙ ˙ ˙ i ˙ ˙ ˙ j ˙ ˙ ˙ n & \text{for}1\le i<j\le n\text{.}\end{array}$$ $$C_0=\{\lambda ={\lambda }_1{\epsilon }_1+\dots +{\lambda }_n{\epsilon }_n\mid {\lambda }_i\in ℝ,{\lambda }_1\ge \dots \ge {\lambda }_n\}$$

Coxeter's theorem Let $({𝔥}_{ℝ},W_0)$ be a Euclidean refelction group. Let $C_0$ be a fundamental region for $W_0$ acting on ${𝔥}_{ℝ}\text{.}$ Let

$$\begin{array}{c}{𝔥}^{{\alpha }_1^{\vee }},\dots ,{𝔥}^{{\alpha }_n^{\vee }}\text{be the walls of}{C}_0\\ s_1,\dots ,s_n\text{the corresponding reflections}\end{array}$$

Then $W_0$ is presented by generators $s_1,\dots ,s_n$ with reflections

$$s_i^2=1\text{and}\underset{\underset{m_{ij}\text{factors}}{⏟}}{s_i{s}_j{s}_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{s_j{s}_i{s}_j\dots }\text{for}i\ne j$$

where $\raisebox{1ex}{$\pi $}\!\left/ \!\raisebox{-1ex}{$m_{ij}$}\right.={𝔥}^{{\alpha }_i^{\vee }}\measuredangle {𝔥}^{{\alpha }_j^{\vee }}\text{.}$

Groups of Lie Type

Type $G{L}_n$ $G{L}_n\left(ℂ\right)$ is generated by

$$\begin{array}{c}{x}_{ij}\left(c\right)=\left(\begin{array}{ccc}1& & c\\ & \ddots & \\ & & 1\end{array}\right),x_{ji}\left(c\right)=\begin{array}{cc}& i\\ j& \left(\begin{array}{ccc}1& & \\ & \ddots & \\ c& & 1\end{array}\right)\end{array}\\ h_{\lambda }\left(t\right)=\left(\begin{array}{ccc}{t}^{{\lambda }_1}& & 0\\ & \ddots & \\ 0& & t^{{\lambda }_n}\end{array}\right)\text{for}\begin{array}{c}1\le i<j\le n,c\in ℂ,\\ \lambda ={\lambda }_1{\epsilon }_1+\dots +{\lambda }_n{\epsilon }_n\in {𝔥}_{\mathbb{Z}},t\in {ℂ}^{\times }\end{array}\end{array}$$

with relations

$$\begin{array}{c}{x}_{ij}\left(c_1\right)x_{ij}\left(c_2\right)=x_{ij}(c_1+c_2),\dots \\ w{x}_{ij}\left(c\right)w^{-1}=x_{w\left(i\right)w\left(j\right)}\left(c\right),w{h}_{\lambda }\left(t\right)w^{-1}=h{w}_{\lambda }\left(t\right)\\ \multicolumn{2}{c}{\text{and more...}}\end{array}$$

Type $S{L}_3$ $S{L}_3\left(𝔽\right)$ is generated by

$$\begin{array}{ccc}{x}_{{\alpha }_1^{\vee }}\left(c\right)=\left(\begin{array}{cc}1& c\\ & 1\\ & & 1\end{array}\right)& x_{{\alpha }_2^{\vee }}\left(c\right)=\left(\begin{array}{c}1\\ & 1c\\ & & 1\end{array}\right)& x_{{\alpha }_3^{\vee }}\left(c\right)=\left(\begin{array}{ccc}1& & c\\ & 1\\ & & 1\end{array}\right)\\ x_{-{\alpha }_1^{\vee }}\left(c\right)=\left(\begin{array}{cc}1& \\ c& 1\\ & & 1\end{array}\right)& x_{-{\alpha }_2^{\vee }}\left(c\right)=\left(\begin{array}{c}1\\ & 1\\ & c& 1\end{array}\right)& x_{-{\alpha }_3^{\vee }}\left(c\right)=\left(\begin{array}{ccc}1& & \\ & 1\\ c& & 1\end{array}\right)\\ h_{{\alpha }_1}\left(t\right)=\left(\begin{array}{c}t\\ & t^{-1}\\ & & 1\end{array}\right)& h_{{\alpha }_2}\left(t\right)=\left(\begin{array}{c}1\\ & t\\ & & t^{-1}\end{array}\right)\end{array}$$

with relations

$$x_{{\alpha }_1^{\vee }}\left(c_1\right)x_{{\alpha }_1^{\vee }}\left(c_2\right)=x_{{\alpha }_1^{\vee }}(c_1+c_2),h_{{\alpha }_1}\left(t_1\right)h_{{\alpha }_1}\left(t_2\right)=h_{{\alpha }_1}\left(t_1{t}_2\right),\text{etc.}$$

(Chevalley-Steinberg-Tits) Let $({𝔥}_{\mathbb{Z}},W_0)$ be a crystallographic reflection group.

$R^{+}$ an index set for the reflections in $W_0\text{.}$

Define $G$ by generators

$$x_{\alpha }\left(c\right),x_{-\alpha }\left(c\right)\text{and}{h}_{\lambda }\left(t\right),\text{for}\begin{array}{c}\alpha \in R^{+},c\in ℂ\\ \lambda \in {𝔥}_{\mathbb{Z}},t\in {ℂ}^{\times }\end{array}$$

with relations

$$\begin{array}{c}{x}_{\alpha }\left(c_1\right)x_{\alpha }\left(c_2\right)=x_{\alpha }(c_1+c_2),\\ h_{\lambda }\left(t_1\right)h_{\lambda }\left(t_2\right)=h_{\lambda }\left(t_1{t}_2\right),h_{\lambda }\left(t\right)h_{\mu }\left(t\right)=h_{\lambda +\mu }\left(t\right),\\ \multicolumn{2}{c}{\text{and more.}}\end{array}$$

Then

$$\begin{array}{ccc}\left\{\begin{array}{c}\mathbb{Z}\text{-reflection groups}\\ ({𝔥}_{\mathbb{Z}},W_0)\end{array}\right\}& ⟷& \left\{\begin{array}{c}\text{complex reductive}\\ \text{algebraic groups}\end{array}\right\}\end{array}$$

is an equivalence of categories

Other equivalences

$$\begin{array}{ccc}\left\{\begin{array}{c}\text{complex reductive}\\ \text{algebraic groups}\end{array}\right\}& ⟷& \left\{\begin{array}{c}\text{compact Lie}\\ \text{groups}\end{array}\right\}\end{array}$$ $$\begin{array}{ccc}\left\{\begin{array}{c}\text{connected, simply}\\ \text{connected, compact}\\ \text{Lie groups}\end{array}\right\}& ⟷& \left\{\begin{array}{c}\text{complex semisimple}\\ \text{Lie algebras}\end{array}\right\}\end{array}$$

Anderson-Grodal et al have proved:

There is an equivalence of categories:

$$\begin{array}{ccc}\left\{{\mathbb{Z}}_q\text{-reflection groups}\right\}& ⟶& \left\{p\text{-compact groups}\right\}\\ ({𝔥}_{{\mathbb{Z}}_q},W_0)& ⟼& BG\end{array}$$

Notes and References

These are a typed version of lecture notes for the first in a series of lectures given at the Brazil Algebra Conference, Salvador, 16 July 2012.

page history