Lectures in Representation Theory

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

Last update: 12 August 2013

Lecture 2

Homework Problem 1.16 Study L2(G) for large groups (the algebra of L2 functions on G). Go find out about compact groups and representation theory.

If V is a (finite dimensional) A-module of dimension d, then there is an associated representation, which we shall also denote by V, such that V:AMd().

Definition 1.17 Let A be an algebra, and V1 and V2 be A-modules. The direct sum V1V2 is the A-module of all pairs (v1,v2), where viVi. Addition is defined componentwise; the action of A is a·(v1,v2)= (av1,av2).

We can make a parallel definition for representations.

Definition 1.18 Let A be an algebra, and V1:AMd1() and V2:AMd2() representations of A. Then the direct sum (V1V2):AMd1+d2() is the representation given by

(V1V2)(a)= ( V1(a)0 0V2(a) ) .

Definition 1.19 An A-module V is completely decomposable if it is isomorphic (as an A-module) to a direct sum of simple A-modules.

Problem 1.20 Let A be an algebra.

1. What are the irreducible representations of A?
2. Are the other representations of A completely decomposable?

Definitions 1.21 Given an algebra A. A trace on A is a linear functional t:A such that t(ab)=t(ba), for all a,bA. The center of A is the subalgebra Z(A)={tA|ta=ataA}. An idempotent is an element pA such that p0 and p2=p. A central idempotent is an idempotent z such that zZ(A). An ideal of A is an ideal in the ring theory sense.

We will use the words “trace” and “character” interchangeably; some would have us call our traces “virtual characters”.

Definition 1.22 Let V:AMd() be a representation. Then the character of the representation V is given by χV:A: atr(V(a)), where tr denotes the vanilla trace of high school matrix theory. Any such character is a trace in the sense defined above.

The following propositions can be nicely summarized as follows:

Theorem 1.23 There exists a unique nonzero

{ trace element of the center central idempotent ideal irreducible representation } inMd() { up to scalar multiples up to scalar multiples up to equivalence } .

Proposition 1.24 There is a unique (up to scalar multiples) nonzero trace on Md().


Let t be a trace, and let aMd().

t(a) = t ( ij aijEij ) = ijaij t(Eij)= ijaij t(Ei1E1j) = ijaijt (E1jEi1)= ijaij δijt(E11) =t(E11)tr(a)

Thus any such t is a scalar multiple of the ordinary trace functional.

Proposition 1.25 There is a unique nonzero element in Z(Md()) (up to scalar multiples).


Let zZ(Md()), z0. Now z=ijzijEij, and z0 implies that for some m and n the coefficient zmn0, Fix these m and n.

Let ks. Since z is central, EmkzEsn= EmkEsnz= 0. On the other hand,

EmkzEsn = Emk ( ijzij Eij ) Esn = Emkzks EksEsn = zksEmn

Thus if ks, then zks=0. Thus z central implies that z is a diagonal matrix.

To show that z central implies z is scalar, fix 1id, and consider Eiiz. For any j, Eiiz= EijEjiz= EijzEji= zjjEii. Since this holds for all j, it follows that zjj=z11 for all j.

Proposition 1.26 There is a unique nonzero ideal in Md().



Proposition 1.27 There is a unique central idempotent in Md().


Let zZ(Md()) with z2=z. z central implies that z=z11Id. Now z2=(z11Id)2=z112Id; thus z11 is a complex number satisfying z112=z11, so either z11=0 or z11=1. This is one place where these developments will not go through so nicely in other characteristics.

Proposition 1.28 There is a unique (nonzero) irreducible representation of Md() (up to equivalence).


First we shall prove that such a representation exists.

Let V be the space of all column vectors of length d, d. Md() acts on V by matrix multiplication.

Claim: V is irreducible.

Let e1,,ed be the standard basis for V. Let W be a subrepresentation, with W0. Choose vW with v0, v=viei, and fix an i such that vi0.

Claim: Md()v=V (whence W=V, proving the previous claim.)

1viEjiv = 1viEji kvkek = 1vikvk Ejiek = 1vikvk δikej = 1viviej=ej

Thus for every j, ejMd()v. Thus Md()v=V.

Continued in next lecture.

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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