Some Commutant Algebras
Next, let and be two inequivalent matrix irreps, with degrees and , respectively, and consider the representation . As a matrix, this looks like:
Where we’ve broken the rows and columns into blocks of size and . Now let’s determine the algebra of matrices commuting with each such matrix . Let’s break down into blocks like .
The nice thing about this is that when the block sizes are the same, and when we break rows and columns into the same blocks, the rules for multiplication are the same as for regular matrices:
If these are to be equal, we have four equations to satisfy:
And we can apply Schur’s lemma to all of them. In the middle two equations, we see that both and must be either be invertible or zero. But if either one is invertible, then it gives an equivalence between the matrix irreps and . But since we assumed that these are inequivalent, we conclude that and are both the appropriate zero matrices. And then the first and last equations are handled just like single irreps were last time. Thus we must have
And so , where the multiplication is handled component by component. Similarly, the direct sum of pairwise-inequivalent irreps has commutant algebra , with multiplication handled componentwise. The degree of the representation is the sum of the degrees of the irreps, and the dimension of the commutant is .