Last update: 27 August 2013
More generally, let be a finite index set and suppose where for each Let and let
Elements of are matrices of the form where is a matrix with complex entries. So is a subalgebra of consisting of block diagonal matrices. Note that we do not insist that so is merely block diagonal and not necessarily semisimple.
Proposition 3.13 The centralizer of an algebra of block diagonal matrices is the direct sum of the centralizers of each of the blocks. That is,
Since block diagonal matrices multiply componentwise, it is obvious that Suppose, on the other hand, that
Elements of may be written as
commutes with all such elements, hence
Multiplying out, we obtain
Comparing coefficients of on each side, we obtain
for all and for all choices of elements
Corollary 3.15 If is a finite index set, then
Let be a representation of an arbitrary (associative) algebra and suppose is completely decomposable. Denote the decomposition of by where the denote, as usual, representatives of distinct isomorphism classes of irreducible Let
If we choose a basis for each irreducible then we may choose a basis for by taking unions of copies of each With respect to this basis, the matrices of are of the form where there are many blocks of for each Thus, in each matrix each block consists of the same matrix repeated times, and so as algebras.
From the previous propositions, So the block of a matrix in the centralizer consists of scalar matrices That is, for
We next perform a (somewhat complicated) reordering of the basis of the module In terms of representations, this exchanges for the equivalent representation where is a product of elementary matrices describing the permutation of the basis.
Recall that interchanging two columns and of a matrix is equivalent to multiplying on the right by the elementary matrix Multiplying by this elementary matrix on the right interchanges rows and Note that this matrix is self-invertible, and so conjugating a matrix by (which is equivalent to interchanging the and basis vectors) corresponds to interchanging the and rows and columns of We may thus describe the permutation of the basis and the resulting change on and is to similarly permute their rows and columns.
We number the columns of blocks, starting from the left, from 1 to There are columns within a particular column of blocks, which we label 1 up to from left to right. Hence, the column of block of the component may be labeled by the ordered pair Since the individual blocks are square, we may label the rows similarly using the number of their block and row within that block. Thus ranges from 1 to and ranges from 1 to and the entry of the component is
Permute the columns so that they appear in the order (i.e. interchange the block number and column number). After permuting the columns, but before performing the similar permutation of the rows, the component of is of the form
It should be noted that after rearranging columns, the obvious block structure no longer consists of square blocks. Each block above is a by matrix. Therefore, the rows of the component are still labeled by with and
If we now perform the permutation of the rows so that they are ordered observe that we will collect all of the rows with nonzero entries in the first column of blocks into the first rows. In fact, the component becomes where is an matrix.
Since is block diagonal, we may perform this reordering simultaneously on each component. Hence, after this reordering of the basis, we may regard We have shown that but for any algebra
Proposition 3.17 Let be an arbitrary associative algebra and let be a completely decomposable finite dimensional representation of If is a decomposition of into pairwise nonisomorphic irreducible modules, then where
In particular, is semisimple and where is the multiplicity of in
We have fully described 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 and would like to simplify the to matrices.
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.