Centers of Commutant Algebras
We want to calculate the centers of commutant algebras. We will have use of the two easily-established equations:
Where , , , and are linear functions. In particular, this holds where and are matrices representing linear endomorphisms of , and and are matrices representing linear endomorphisms of .
Now let be a matrix representation and consider a central matrix . That is, for all , we have
Let’s further assume that we can write
where each is an irreducible representation of degree . Then we know that we can write
Thus we calculate:
This is only possible if for each we have for all . But this means that is in the center of , which implies that . Therefore a central element can be written
As a concrete example, let’s say that , where and . Then the matrices in the commutant algebra look like:
and the dimension of the commutant algebra is evidently .
The central matrices in the commutant algebra, on the other hand, look like:
And the dimension is .