Last update: 19 August 2013
We would like to show that the basis of matrix units of is contained in whence that We need to produce scalars such that
By analogy with the way the matrix units arise in the case of the regular representation (do either of you want me to type that up as well?), wherein the dual are constructed using the ordinary trace tr, one may hope that will hold in this more general situation. In fact,
Thus we have shown that all of the matrix units of are in that is and thus
Homework Problem 1.50 Where in this proof that the nondegeneracy of implies that is isomorphic to a direct sum of matrix rings does the Jacobson density theorem occur?
Homework Problem 1.51 Prove the converse: that if is isomorphic to a direct sum of matrix rings, then the trace of the regular representation is nondegenerate.
Definition 1.52 Let be a faithful representation of and let be the trace of The radical of is the set of elements of at which the trace is degenerate; that is,
A priori, the definition of the radical would seem to be dependent on the choice of nondegenerate representation this turns out to not be the case, as we shall see below.
Theorem 1.53 Given a faithful representation of and the associated trace and radical,
|1.||is an ideal of|
|2.||If is an ideal of nilpotent elements, then (Recall that an element is nilpotent provided for some positive integer|
|3.||Every element of is nilpotent.|
1: Let and Then for every
The second equality holds since is a trace; the last since was chosen to be in the radical.) Thus is invariant under right multiplication from Similar arguments show that is invariant under left multiplication, and closed under addition.
2: Let be an ideal of nilpotent elements, and let To show that consider for any Since is an ideal, we have whence is nilpotent. Thus is a nilpotent matrix in and so its trace Thus and so as desired.
3: Given an element show that is nilpotent. Note that for every we have Thus we have that the trace for all We want to show that this implies that the matrix is nilpotent.
I can’t think without symmetric functions. – A.Ram
We know that is similar to an upper triangular matrix, and what the powers of such a matrix are:
Thus for every we know that
One has the following Newton identity (c.f. Macdonald Symmetric functions and Hall polynomials):
For our current purposes, the point is that for all and therefore all of the are zero.
Now consider the characteristic polynomial of
Thus the characteristic polynomial of the matrix is and so we have that
Once is nilpotent, we know that a is nilpotent as well.
Corollary 1.54 The radical is the largest ideal of nilpotent elements in whence is independent of the choice of the faithful representation
Corollary 1.55 The trace of the regular representation is nondegenerate if and only if
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.