Lectures in Representation Theory

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

Last update: 19 August 2013

Lecture 6

Continued proof.

Let e=giGgigi*; we wish to show that e=1.

Let aA be arbitrary, and denote the coefficient of gi in any element a by a|gi. Then tr(ae)= giG tr(agigi*)= giG ag,g*= giG (agi)|gi= tr(a) using the dual basis property. Hence tr(a(1-e))=0 for all aA, and the nondegeneracy of tr implies 1-e=0, that is e=1.

Hence P1:VV1 is a projection.

Furthermore, V2=(I-P1)V is an A-module. Indeed, by the lemma from last time, A commutes with P1, hence with I-P1. Therefore, for any aA and vV, a·(I-P1)v=(I-P1)a·vV2, hence I-P1 is an A-module homomorphism mapping V onto V2.

Finally, we assert that V=V1V2. Any vV may be written v=P1v+(I-P1)v, hence V=V1+V2. To show the sum is direct, note first that P12=P1 since P1 fixes V1=imP1 pointwise. Suppose xV1V2, so we may write x=(I-P1)v for some vV. Then P1x=x as xV1, and hence x=P1x=P1 (I-P1)v=P1 v-P12v=0. which completes the proof of (1)(2).

Next, prove (1)(3). By the previous case, we may also assume that the all finite dimensional modules are completely decomposable.

Let V=A be the hideously denoted left regular representation of A. Since dimA<, we may write VλVˆ(Wλ)mλ, where Vˆ is some (finite) index set.

Note that the representation V:AMd() is injective (sometimes called faithful) as V(a)·1=a for all aA. Thus AV(A)λVˆ(Wλ(A))mλ as algebras. The number mλ are called the multiplicity of the irreducible representation Wλ in V(A)A.

For convenience, write W=λVˆ(Wλ(A))mλ. Note that an arbitrary element of W is of the form W(a)= ( Wλ(a) 0 Wλ(a) Wμ(a) 0 Wν(a) ) for some aA, where there are mγ occurrences of each Wγ(a) in the block diagonal decomposition of V(a). The product in the algebra W(A) is componentwise, hence the map WλVˆWλ given by deleting duplicate copies of irreducibles, i.e. W(a)λVˆWλ(a) is an (onto) algebra homomorphism.

It is not hard to see that these algebras are isomorphic. Let {g1λ,g2λ,,gdλλ} be a basis for Wλ, and let gi,jλW be the corresponding element to giλ in the jth copy of Wλ in W. The set { hiλ= j=1dλ gi,jλ| 1idλ, λVˆ } is a basis for W, since an element of W must have identical matrices in all blocks indexed by a given λ. Hence W(A) has the same dimension as λVˆWλ(A); indeed, the homomorphism above sends hiλ to giλ (following the same indexing scheme) and is clearly invertible.

Setting Aˆ=Vˆ, we have AV(A)λAˆWλ(A). It remains to show that Wλ(A)Mdλ(). This follows from the next extremely useful lemma:

Lemma 1.49 (Schur) Suppose V and W are irreducible representations of A of dimensions d1 and d2, respectively. If B is a d1×d2 matrix such that V(a)B=BW(a) for all aA, then either B=0 or V is equivalent to W and B=cI.


Let V=d1 and W=d2 represent the corresponding A-modules of these representations, and assume B0. Let {w1,,wd2} be the standard basis of W, so that B represents a linear transformation WV given by vi=Bwi. The condition V(a)B=BW(a) becomes B(a·w)=a·B(w), i.e. B is an A-module homomorphism.

Since B0, the simplicity of W forces kerB=0. Therefore B is an isomorphism of modules, hence d1=d2 and the original representations are equivalent.

Observe that for all c, V(a)(B-cI)=(B-cI)W(a), so B-cI is either zero or an isomorphism. Let c to be an eigenvalue of B (this requires that our field be algebraically closed), hence B-cI is not invertible. Then B-cI=0 or B=cI.

We have Wλ(A)Mdλ(); to show that Wλ(A)=Mdλ(), we will show that Wλ(A) contains the basis of matrix units. Note that giGWλ(g*)Ei,mWλ(g) commutes with Wλ(A) by the lemma of last time. By Schur’s lemma (with V=W=Wλ), this element must be of the form cIdλ for some c (possibly zero). We calculate traces; using the trace property:

tr ( gGWλ (g*)Ei,m Wλ(g) ) = tr ( gGWλ (g*)Wλ (g)Ei,m ) = tr ( Wλ (giGgi*gi1) Ei,m ) =tr(Ei,m) = δi,mdλ.

On the other hand, tr(cIdλ)=cdλ, so c=0 for im and c=(1/dλ)2 if i=m.

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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