Lectures in Representation Theory

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

Last update: 27 August 2013

Lecture 20

More generally, let Vˆ be a finite index set and suppose AλMdλ(), where dλ for each λVˆ. Let d=λdλ and let A=λVˆAλ.

Elements of A are matrices of the form ( aλ aμ 0 0 aδ ) where aρAρ is a dρ×dρ matrix with complex entries. So A is a subalgebra of Md() consisting of block diagonal matrices. Note that we do not insist that Aλ=Mdλ(), so A is merely block diagonal and not necessarily semisimple.

Proposition 3.13 The centralizer of an algebra A of block diagonal matrices is the direct sum of the centralizers of each of the blocks. That is, λVˆAλ= λVˆ Aλ.


Since block diagonal matrices multiply componentwise, it is obvious that (λVˆAλ) λVˆAλ. Suppose, on the other hand, that B= μ,νVˆ Eμ,νbμ,ν A.

Elements of A may be written as λVˆEλ,λaλ where aλAλ. Then B commutes with all such elements, hence (λVˆEλ,λaλ) (μ,νVˆEμ,νbμ,ν) = (μ,νVˆEμ,νbμ,ν) (λVˆEλ,λaλ) . Multiplying out, we obtain λ,νVˆ Eλ,νaλ bλ,ν= μ,λVˆ Eμ,λ bμ,λaλ. Comparing coefficients of Eλ,ν on each side, we obtain aλvλ,ν= bλ,νaν (3.14) for all λ,νVˆ and for all choices of elements aμAμ.
Taking μ=λ we obtain aλbλ,λ= bλ,λaλ for all aλAλ; hence, bλ,λAλ for all λVˆ.
Fix λVˆ and choose aλ=I and aμ=0 for μλ. Then 3.14 becomes bλ,ν= aλbλ,ν =bλ,νaν =0 so bλ,ν=0 whenever λν.
Thus B=μVˆEμ,μbμ where bμAμ. Therefore, AλVˆAλ.

Corollary 3.15 If Aˆ is a finite index set, then λAˆMdλ()= λAˆ Mdλ() =λAˆ Idλ

Let V be a representation of an arbitrary (associative) algebra A and suppose V is completely decomposable. Denote the decomposition of V by VλVˆ (Vλ)mλ =λVˆmλ Vλ where the Vλ denote, as usual, representatives of distinct isomorphism classes of irreducible A-modules. Let dλ=dimVλ.

If we choose a basis λ={w1λ,w2λ,wdλλ} for each irreducible Vλ, then we may choose a basis for V by taking unions of mλ copies of each λ. With respect to this basis, the matrices of V(A) are of the form V(a)=λVˆ ( Vλ(a) Vλ(a) 0 0 Vλ(a) ) where there are mλ many blocks of Vλ for each λ. Thus, in each matrix V(a), each λ block consists of the same matrix repeated mλ times, and so V(A)λ Imλ(Vλ(A)) (3.16) as algebras.

From the previous propositions, V(A) = λVˆ Imλ (Vλ(A)) = λVˆ Imλ (Vλ(A)) = λVˆ Mmλ (Vλ(A)) = λVˆ Mmλ (Idλ()) So the λ block of a matrix b in the centralizer V(A) consists of mλ2 scalar matrices ci,jλIdλ. That is, for bV(a), b=λVˆ ( c1,1λ c1,1λ c1,1λ c1,mλλ c1,mλλ c1,mλλ cmλ,1λ cmλ,1λ cmλ,1λ cmλ,mλλ cmλ,mλλ cmλ,mλλ )

We next perform a (somewhat complicated) reordering of the basis of the module V. In terms of representations, this exchanges V for the equivalent representation aP-1V(a)P where P is a product of elementary matrices describing the permutation of the basis.

Recall that interchanging two columns i and j of a matrix is equivalent to multiplying on the right by the elementary matrix E=ki,jEk,k+Ei,j+Ej,i. Multiplying by this elementary matrix on the right interchanges rows i and j. Note that this matrix is self-invertible, E=E-1, and so conjugating a matrix X by E (which is equivalent to interchanging the ith and jth basis vectors) corresponds to interchanging the ith and jth rows and columns of X. We may thus describe the permutation of the basis and the resulting change on V(A) and V(A) is to similarly permute their rows and columns.

We number the columns of blocks, starting from the left, from 1 to mλ. There are dλ columns within a particular column of blocks, which we label 1 up to dλ from left to right. Hence, the nth column of mth block of the λ component may be labeled by the ordered pair (m,n). Since the individual blocks are square, we may label the rows similarly using the number of their block and row within that block. Thus m ranges from 1 to mλ and n ranges from 1 to dλ and the {(m1,n1),(m2,n2)} entry of the λ component is δn1,n2cm1,m2λ.

Permute the columns so that they appear in the order (1,1), (2,1), (3,1), , (mλ,1), (1,2), (2,2), , (mλ,dλ) (i.e. interchange the block number and column number). After permuting the columns, but before performing the similar permutation of the rows, the λ component of b is of the form ( c1,1 c1,2 c1,mλ c1,1 c1,2 c1,mλ c1,1 c1,2 c1,mλ c2,1 c2,2 c2,mλ c2,1 c2,2 c2,mλ c2,1 c2,2 c2,mλ cmλ,1 cmλ,2 cmλ,mλ cmλ,1 cmλ,2 cmλ,mλ cmλ,1 cmλ,2 cmλ,mλ )

It should be noted that after rearranging columns, the obvious block structure no longer consists of square blocks. Each block above is a dλ by mλ matrix. Therefore, the rows of the λ component are still labeled by (m,n) with 1mmλ and 1ndλ.

If we now perform the permutation of the rows so that they are ordered (1,1), (2,1), (3,1), , (dλ,mλ), observe that we will collect all of the rows with nonzero entries in the first column of blocks into the first mλ rows. In fact, the λ component becomes ( Cλ Cλ Cλ ) =IdλCλ where Cλ=(ci,jλ) is an mλ×mλ matrix.

Since b is block diagonal, we may perform this reordering simultaneously on each component. Hence, after this reordering of the basis, we may regard b=λIdλCλ. We have shown that V(a)λIdλ(Mmλ()), but for any algebra A, Idλ(A)A.

Proposition 3.17 Let A be an arbitrary associative algebra and let V be a completely decomposable finite dimensional representation of A. If V=λmλVλ is a decomposition of V into pairwise nonisomorphic irreducible modules, then V(A) λVˆ Mmλ() where dλ=dimVλ.

In particular, V(A) is semisimple and dimV(A)=λmλ2, where mλ is the multiplicity of Vλ in V.

We have fully described V(A) in terms of complex matrices, which already yields the useful information that centralizer algebras of arbitrary finite dimensional representations are semisimple. In 3.16, we showed that V(A)=λVˆImλ(Vλ(A)) and would like to simplify the Vλ(A) to matrices.

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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