Representation Theory, Reflection groups and Groups of Lie Type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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).

A simple A-module is an A-module M with no submodules, except D and M.

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=NP.


Reflection groups

Let Z be a subring of .

A reflection group is a pair (𝔥Z,W0) with

  1. 𝔥Z a free Z-module
  2. W0 a finite subgroup of GL(𝔥Z) generated by reflections.

A reflection is an element sGLn() conjugate to

( ξ0 1 01 ) withξ1.

A crystallographic reflection group is a -reflection group.

A Euclidean reflection group is an -reflection group.


Type SL3: 𝔥=-span{α1,α2}

W0= s1s2 si2=1,s1 s2s1=s2s1 s2 𝔥α1 𝔥α2 𝔥α3 s1 s2 α2 α1 s1s2 s2s1 s1s2s1=s2s1s2 C0

Type GLn: 𝔥=ε1 ++εn.

W0=Snpermuting ε1,,εn

Reflections in Sn:

sij= 1 ˙ ˙ ˙ i ˙ ˙ ˙ j ˙ ˙ ˙ n for1i<jn. C0= { λ=λ1ε1++ λnεn λi,λ1 λn }

Coxeter's theorem Let (𝔥,W0) be a Euclidean refelction group. Let C0 be a fundamental region for W0 acting on 𝔥. Let

𝔥α1,, 𝔥αn be the walls ofC0 s1,,sn the corresponding reflections

Then W0 is presented by generators s1,,sn with reflections

si2=1and sisjsimijfactors= sjsisjmijfactors forij

where πmij= 𝔥αi 𝔥αj.

Groups of Lie Type

Type GLn GLn() is generated by

xij(c)= ( 1 c 1 ) ,xji(c)= i j ( 1 c 1 ) hλ(t)= ( tλ1 0 0 tλn ) for 1i<jn,c, λ=λ1ε1++ λnεn𝔥, t×

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 SL3 SL3(𝔽) 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 (𝔥,W0) be a crystallographic reflection group.

R+ an index set for the reflections in W0.

Define G by generators

xα(c), x-α(c)and hλ(t), for αR+,c λ𝔥,t×

with relations

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


{ -reflection groups (𝔥,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:

{q-reflection groups} {p-compact groups} (𝔥q,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.

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