Lectures in Representation Theory

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

Last update: 12 August 2013

Lecture 3

Continued proof.

Having shown that V is an irreducible irreducible representation of Md(), we proceed to prove uniqueness of V up to equivalence.

Let W be any simple module of Md() and choose wW\{0}. Then

0w=Idw=j=1d Ej,jw

and so Ei,iw0 for some i. The set Md()Ei,iwW is a nonzero left submodule of W and hence W=Md()Ei,iw by the simplicity of W.

We claim that VMd()Ei,i as left Md()-modules. The map defined by

(a1a2ad) ( 0 a1 0 a2 ad )

realizes this isomorphism, where the nonzero entries on the right appear in the ith column (the proof is left as an exercise).

We may then define a linear transformation

ϕ:VMd() Ei,iMd() Ei,iw=W

by ϕ(ej)= Ej,iEi,iw= Ej,iw. This map is a module homomorphism, for if we choose an arbitrary matrix unit Ek,, then

ϕ(Ek,ej)= ϕ(δ,jek)= δ,jEk,iw= Ek,jEj,iw= Ek,ϕ(ej)

and the module property follows by linearity.

Since kerϕV is a submodule and eikerϕ, the simplicity of V forces kerϕ=0. Similarly, imϕ is a nonzero submodule of the simple module W, hence imϕ=W. The map ϕ is therefore an isomorphism between V and W.

Next, let us fix some notation. As usual, let be the complex numbers, the reals, the integers. We depart from standard notation to write ={0,1,2,} for the set of nonnegative integers and let ={0,1,2,} denote the set of positive integers. We will use interval notation [1,n]={1,2,,n} to denote the set of integers between two endpoints (inclusive).

Definition 1.29 A simple algebra A is an algebra with no nonzero proper two-sided ideals. For this course, a simple algebra is any algebra A such that AMd() for some d.

Definition 1.30 We will say a finite dimensional algebra A is semisimple provided AλAˆMdλ(), where Aˆ is some finite index set, and dλ.

We will often write AλMdλ() for the simple component of A indexed by λAˆ. Thus a typical element aA is of the form

a= ( aλ1 aλ2 0 0 aλk )

where each aλjAλjMdλj(), which we regard as a dλj×dλj matrix.

Every finite dimensional semisimple algebra has a basis of matrix units {Ei,jλ|1i,jdλ,λAˆ}, which correspond to the usual matrix units in λAˆMdλ(). That is, the structure constants are given by

Ei,jλ Er,sμ= δλ,μ δj,r Ei,sλ.

Define a trace χλ:A by χλ(a)=tr(aλ)=i=1dλai,iλ where a=i,j,λai,jλEi,jλ. That is χλ is the trace of the λ block of a.

Proposition 1.31 Let t be a trace on A and let tλ=t(E1,1λ). Then

t=λAˆ tλχλ.


By 1.23, tr is the unique trace on Md(), hence χλ restricted to Aλ is the unique trace on Aλ. Thus, for each λAˆ, there exists tλ such that t(aλ)=tλχλ(aλ) for all aλAλ. In particular,

t(E1,1λ) =tλχ(E1,1λ) =tλtr(E1,1) =tλ.

The proposition follows by linearity of t.

Definition 1.32 Let t be a trace on A. The vector (tλ)λAˆ weight vector of t.

The previous proposition shows that a trace t is completely determined by its weight vector. Conversely, one may define a trace by specifying its weight vector. Hence the trace functionals on A are in one to one correspondence with the vectors in |Aˆ| which explains the notation t.

After classifying the traces on A, we turn to the ideal structure. From the general theory of semisimple algebras, we know that A possesses a finite collection of minimal two-sided ideals (i.e. nonzero ideals of A containing no proper nonzero two-sided ideals). The algebra A is the direct sum of these minimal ideals and, furthermore, every ideal of A is a direct sum of some subset of these ideals.

Since Md() is simple and AλAˆMdλ(), evidently the minimal two-sided ideals of A are the simple components AλMdλ(), λAˆ.

Definitions 1.33 Two idempotents z1 and z2 are said to be orthogonal provided z1z2=z2z1=0. An idempotent z is said to be minimal (or principal) provided z cannot be written as a sum of orthogonal idempotents.

From the general theory, every minimal two-sided ideal Aλ is generated by a central idempotent zλ, i.e. Aλ=Azλ. Since zλ must be the identity of Aλ, necessarily zλIdλ=i=1dλEi,iλ. Note that the zλ are orthogonal as they belong to different minimal ideals of A.

However, the minimality of the two-sided ideals Aλ does not quite translate into minimality of the zλ in the sense of the above definition, since the Ei,iλ are orthogonal idempotents which sum to zλ. Yet zλ is minimal in the sense that it may not be written as the sum of central orthogonal idempotents:

Proposition 1.34 The minimal central idempotents of A are {zλ|λAˆ}. In particular, this set forms a basis for the center Z(A).


One may verify by using the basis of matrix units that Z(Md())=Id. Since A is a direct sum of matrix algebras, the second statement of the proposition follows from the definition of the zλ.

Suppose zZ(A) is any central idempotent of A. If we express z=λAˆcλzλ, cλ, in terms of the new basis for the center, we obtain

zλ=zλ2= (λAˆcλzλ)2 =λAˆ cλ2zλ

using the orthogonality of the zλ. By uniqueness of expression, we have cλ2=cλ and hence cλ{0,1}. Thus, any central idempotent of A is a sum of one or more zλ. The result then follows from the independence of the zλ.

Homework Problem 1.35 Show that every element of the form

Ei,iλ+ ji ci,j Ei,jλ,

where each ci,j{0,1} is an idempotent which generates a minimal left ideal of A. Conclude that these elements are minimal idempotents of A.

Similarly, show that the idempotents

Ei,iλ+ ji ci,jEj,iλ

generate minimal right ideals of A and hence are minimal idempotents.

Finally, we turn to the irreducible representations and simple modules of A. Let Wμ:AMdμ(), μAˆ be defined by Wμ(a)=aμ where aμ=i,j=1dμai,jμEi,jμAμ is the μ-block of the element a.

Proposition 1.36 The representations Wμ, μAˆ are the complete set of irreducible representations of A up to equivalence.


The restriction WμAμ:AμMdμ() is the unique irreducible representation of Aμ. Any A-subrepresentation ϕ:AMdμ() of Wμ is also a Aμ-subrepresentation of Wμ, hence is zero on Aμ. However, kerWμ=λμAλ. Hence, ϕ=0 on A and Wμ is an irreducible representation of A.

If W:AMd() is any irreducible representation of A, then kerW is an ideal of A, and hence is the direct sum of one or more of the Aλ. If Aμ is in the complement of kerW, then Wμ is a summand of W. Irreducibility of W then implies that W is equivalent to exactly one of the Wμ.

Turning to simple modules, denote the space dμ by Wμ, and let A act on Wμ by a·w=aμw. The proof of the following proposition is left as an exercise.

Proposition 1.37 The set Wμ form a complete set of simple modules of A up to isomorphism.

Observe that AEi,jλWλ.

Observe also that the irreducible representation Wλ:AMdλ() defined by a aWλ(a)=aλ has character χWλ:A defined by

χWλ(a)=tr (Wλ(a))=tr (aλ)=χλ.

Hence the irreducible characters of A are the traces χλ, λAˆ.

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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