## Representation Theory, Reflection groups and Groups of Lie Type

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 $M=N\oplus P\text{.}$

$N P N P M=N⊕P 0⟶P⟶M⟶N⟶0 butM≠N⊕P$

## Reflection groups

Let $Z$ be a subring of $ℂ\text{.}$

A reflection group is a pair $\left({𝔥}_{Z},{W}_{0}\right)$ 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

$( ξ0 1 ⋱ 01 ) withξ≠1.$

A crystallographic reflection group is a $ℤ$-reflection group.

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

### Examples

Type $S{L}_{3}:$ ${𝔥}_{ℤ}=ℤ\text{-span}\phantom{\rule{0.2em}{0ex}}\left\{{\alpha }_{1},{\alpha }_{2}\right\}$

$W0= ⟨ s1s2∣ si2=1,s1 s2s1=s2s1 s2 ⟩$ $𝔥α1∨ 𝔥α2∨ 𝔥α3∨ s1 s2 α2 α1 s1s2 s2s1 s1s2s1=s2s1s2 C0$

Type $G{L}_{n}:$ ${𝔥}_{ℤ}=ℤ{\epsilon }_{1}+\dots +ℤ{\epsilon }_{n}\text{.}$

$W0=Snpermuting ε1,…,εn$

Reflections in ${S}_{n}:$

$sij= 1 ˙ ˙ ˙ i ˙ ˙ ˙ j ˙ ˙ ˙ n for1≤i $C0= { λ=λ1ε1+…+ λnεn∣ λi∈ℝ,λ1 ≥…≥λn }$

Coxeter's theorem Let $\left({𝔥}_{ℝ},{W}_{0}\right)$ be a Euclidean refelction group. Let ${C}_{0}$ be a fundamental region for ${W}_{0}$ acting on ${𝔥}_{ℝ}\text{.}$ Let

$𝔥α1∨,…, 𝔥αn∨ be the walls ofC0 s1,…,sn the corresponding reflections$

Then ${W}_{0}$ is presented by generators ${s}_{1},\dots ,{s}_{n}$ with reflections

$si2=1and sisjsi…⏟mijfactors= sjsisj…⏟mijfactors fori≠j$

where $\pi }{{m}_{ij}}={𝔥}^{{\alpha }_{i}^{\vee }}\measuredangle {𝔥}^{{\alpha }_{j}^{\vee }}\text{.}$

## Groups of Lie Type

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

$xij(c)= ( 1 c ⋱ 1 ) ,xji(c)= i j ( 1 ⋱ c 1 ) hλ(t)= ( tλ1 0 ⋱ 0 tλn ) for 1≤i

with relations

$xij(c1) xij(c2)= xij(c1+c2) ,… wxij(c)w-1 =xw(i)w(j) (c),whλ (t)w-1=hwλ (t) and more...$

Type $S{L}_{3}$ $S{L}_{3}\left(𝔽\right)$ is generated by

$xα1∨(c)= ( 1c 1 1 ) xα2∨(c)= ( 1 1c 1 ) xα3∨(c)= ( 1c 1 1 ) x-α1∨(c)= ( 1 c1 1 ) x-α2∨(c)= ( 1 1 c1 ) x-α3∨(c)= ( 1 1 c1 ) hα1(t)= ( t t-1 1 ) hα2(t)= ( 1 t t-1 )$

with relations

$xα1∨(c1) xα1∨(c2)= xα1∨(c1+c2), hα1(t1) hα1(t2)= hα1(t1t2), etc.$

(Chevalley-Steinberg-Tits) Let $\left({𝔥}_{ℤ},{W}_{0}\right)$ be a crystallographic reflection group.

${R}^{+}$ an index set for the reflections in ${W}_{0}\text{.}$

Define $G$ by generators

$xα(c), x-α(c)and hλ(t), for α∈R+,c∈ℂ λ∈𝔥ℤ,t∈ℂ×$

with relations

$xα(c1) xα(c2)= xα(c1+c2), hλ(t1) hλ(t2)= hλ(t1t2), hλ(t)hμ(t)= hλ+μ(t), and more.$

Then

${ ℤ-reflection groups (𝔥ℤ,W0) } ⟷ { complex reductive algebraic groups }$

is an equivalence of categories

Other equivalences

${ complex reductive algebraic groups } ⟷ { compact Lie groups }$ ${ connected, simply connected, compact Lie groups } ⟷ { complex semisimple Lie algebras }$

Anderson-Grodal et al have proved:

There is an equivalence of categories:

${ℤq-reflection groups} ⟶ {p-compact groups} (𝔥ℤq,W0) ⟼ BG$

## 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.