Lectures in Representation Theory

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

Last update: 12 August 2013

Lecture 4

Definition 1.38 An algebra A is semisimple if it is isomorphic to a direct sum

AλAˆ Mdλ()

of matrix algebras where Aˆ is a finite index set and dλ is a positive integer for all λAˆ.

Let A be semisimple algebra, and let ϕ:AλAˆMdλ() be an isomorphism. Then a basis of matrix units for A is

{ eijλ= ϕ-1(Eijλ) |λAˆ,1 i,jdλ } .

For λAˆ, the minimal central idempotent zλ of A indexed by λ is given by

zλ= i=1dλ ei,iλ.

The minimal ideal Iλ of A indexed by λ is given by

Iλ=ϕ-1 (Mdλ()),

and the irreducible representation Wλ of A indexed by λ is given by λAˆ and given by

Wλ: A Mdλ() a ϕ(a)λ

where ϕ(a)λ denotes the λth block of the matrix ϕ(a). In terms of modules, the irreducible module Wλ, is

WλA eijλ.

Since Wλ(a)=ϕ(a)λ for all aA and λAˆ, we let Wijλ(a) denote the i,j-entry of the matrix Wλ(a). Then the irreducible character χλ corresponding to λAˆ is given by

χλ(a)= i=1dλ Wiiλ(a).

Moreover, it should be noted that

Wijλ (ersμ)= δλ,μ δi,r j,s.

Let t be a trace on the semisimple algebra A. Then t is determined by the weight vector (tλ)λAˆ which satisfies the following:

(a) t(a)= λAˆtλχλ(a), for all aA,
(b) t(eiiλ) =tλ.

We suppose that t has the property that tλ0 for all λAˆ (we will see that this means that t is non-degenerate) and define a bilinear form on A by

a,b= t(ab), fora,bA.

The form has the following properties:

(a) symmetric: a,b= t(ab)= t(ba)= b,a,
(b) associative: ab,c= t(abc)= a,bc.
If A has a non-degenerate associative form, then A is a Frobenius algebra. If A has a symmetric, associative non-degenerate form, then A is a symmetric algebra. Many of the results of this chapter hold in these more general algebras.

Let {g1,g2,,gd} be a basis for A. A dual basis with respect to the bilinear form , is the basis {g1*,g2*,,gd*} having the property that

gi,gj* =δij.

The basis {eijλ|λAˆ,1i,jdλ} of matrix units in A has as its dual basis the set {eijλ/tλ|λAˆ,1i,jdλ}. To verify this, we see that

eijλ,esrμ/tμ =tr(eijλesrμtμ)= δλ,μδjstr (eirλtλ) =δλ,μδjs δir.

Theorem 1.39 Let t=(tλ)λAˆ be a trace on A such that tλ0 for all λAˆ.

(a) Fourier Inversion Formula. If λAˆ and 1i,jdλ then eijλ= μAˆ1r,sdμ tλWijλ (esrμtμ) ersμ.
(b) Central Idempotents. If λAˆ then zλ=μAˆ1r,sdμ tλχλ (esrμtμ) ersμ.
(c) Orthogonality of Characters. If λ,νAˆ, then μAˆ1r,sdμ χλ (esrμtμ) χν(ersμ) =δλ,ν dλtλ.


For (a) we have Wijλ(esrμtμ)=δλ,μδj,sδi,r, and the result follows.

For (b), we see that

zλ= i=1dλ eiiλ = i=1dλ μAˆ1r,sdμ tλWiiλ (esrμtμ) ersμ Fourier inversion = μAˆ1r,sdμ tλ i=1dλ Wiiλ (esrμtμ) χλ(esrμtμ) ersμ

Part (c) follows by taking the character χλ of each side the equation in (b).

Our next goal is to show that these formulas are independent of the basis used. To this end we let 𝒜={g1,,gd} and ={h1,,hd} be bases of A and let 𝒜*={g1*,,gd*} and *={h1*,,hd*} be their duals. Let S=(sij) be the transition matrix between 𝒜 and . That is

hi=j=1d sijgj.

If T is the transition matrix between 𝒜* and *, then

δim= hi,hm* = j=1d sijgj, n=1d tmngn* = j,nsij tmn gj,gn* =δj,n = j=1d sijtmj.

Therefore, I=STt.

Now consider the orthogonality of characters formula. We have

jχλ (hj*)χν (hj) = i=1d χλ (ntingn*) χν (msimgm) = m,n ( iti,nsi,m δm,n ) χλ(gn*) χν(gm) = mχλ (gn*) χν(gm).

Thus, orthogonality of characters holds for any basis. In fact, the same argument proves that all three formulas in 1.39 are basis independent. Therefore,

Theorem 1.40 Let A be a semisimple algebra with non-degenerate trace t=(tλ)λAˆ (tλ0), and let 𝒜={g1,,gd} be a basis of A. Then

(a) Fourier Inversion Formula. The elements eijλ= g𝒜tλ Wijλ(g*)g form a complete set of matrix units for A.
(b) Central Idempotents. The central idempotents of A are given by zλ=g𝒜 tλχλ(g*) g.
(c) Orthogonality of Characters. If λ,νAˆ, then g𝒜χλ (g*)χν (g)=δλ,ν dλtλ, where dλ is the dimension of the irreducible A-representation Wλ.

Notes and References

This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.

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