Last update: 19 August 2013
|Proof of in Theorem 1.41.|
Let be the left regular representation of Assuming (2), we have where the are irreducible representations.
Let If then since is a faithful Therefore, for some and there exists so that Then is a non-trivial submodule of the irreducible module so
There exists such that This tells us that is not nilpotent and thus Since is an ideal, and therefore,
Let be a semisimple algebra with basis Let be a non-degenerate trace on (i.e., Let be the dual basis with respect to and for each let
(This is analogous to the sum over all conjugates of a in group theory.) We know that the sum is independent of the basis chosen, so we compute it for the basis of matrix units. Recall that the dual basis is and note that can be written as for some Thus, we have which shows that is a linear combination of the and that
Theorem 1.57 If is a basis of then
Therefore, the set spans Using the Gram-Schmidt process we can orthogonalize this set to compute the minimal central idempotents.
We now specialize this work to the case where is a group algebra. Let be a finite group, and let be its group algebra. We compute the trace of the regular representation for as follows
Therefore, for we have so the dual basis to is This proves that the trace of the regular representation is non-degenerate, and is semisimple. The regular representation of decomposes as Therefore, and so In this case 1.40 reads
The set of all traces forms a space. Since the irreducible characters span the spaces of traces on Since is a group algebra, we have so is constant on the conjugacy classes of That is, is a class function. Let index the conjugacy classes of and for let denote the conjugacy class. Define by Then for some since is a class function. Therefore, the set spans the vector space of traces on and since the are characteristic functions, they are independent and form a basis. Thus, we can index the conjugacy classes of by elements of
Now let denote the value of on and let be the index of the conjugacy class that contains the inverses of the elements of (Note that and may be the same). Then using 1.58, we get Let be the matrix whose rows and columns are indexed by and whose entry in the row and column is and let be the matrix whose rows and columns are indexed by and whose entry in the row and column is Then we have just shown that Moreover, so and we get the second orthogonality relation for finite groups
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.