Last update: 27 August 2013
We saw last time that the Bratteli diagram describes the branching rules for restriction from irreducible modules over to irreducible modules over Note that by Schur’s Lemma and Frobenius Reciprocity, whence that so that the Young lattice also describes branching rules for induction of irreducible
For semisimple, we could produce the irreducible modules by noting that where the were the preimages of the in the block within
Conversely, given an idempotent in we can write where the are minimal idempotents, and with each irreducible. For each minimal idempotent there exists a unique with let us indicate this association by denoting our idempotents as Now if we set one has that and further
Note that as (To prove this, one needs to produce a bijection The appropriate maps are the extensions of and
Now we know that for a minimal idempotent in (and naturally an idempotent of some sort in The branching rule (the induction version) now tells us that Thus if we express the idempotent as a sum of minimal idempotents of we will obtain where the are minimal idempotents in with
Let be a representation of and the associated character, with and We would like to produce a formula for As we know that and by taking traces one then obtains the formula This formula, while interesting, is not necessarily so nice for computational uses. With this in mind, let us define where (as usual) the are the minimal central idempotents of the are the dimensions of the irreducible and the are the multiplicities of these irreducibles. Let be a basis for and be its dual basis with respect to the trace of the regular representation of Now recall that whence that We find then that the values of are encoded in this distinguished element Can we produce an element of within which the values of are similarly encoded?
Let be a basis for let be any nondegenerate trace for and let be the associated dual basis. Define for each
Proposition 2.60 Given the above definitions,
(The second equality is expressing in terms of minimal idempotents of the third is expressing each of those minimal in terms of the minimal idempotents of Note that if we take and symmetrize it, we have Thus we have that
This is a copy of lectures in Representation Theory given by Arun Ram, compiled by Tom Halverson, Rob Leduc and Mark McKinzie.