Product of Representations
Recall that the Frobenius map gives an isomorphism of abelian groups:
Let us compute what the product corresponds to on the RHS.
For that, we take and where and Multiplication gives where is the partition obtained by sorting Next, we use the following standard result.
Lemma. Let act transitively on non-empty set . Pick any and let be its stabiliser. Then:
the trivial module on induced to
The RHS is by definition. Now we map on the LHS to on the RHS. Now:
Since the action on X is transitive, this gives an isomorphism of vector spaces, which clearly commutes with the group action. ♦
In particular is isomorphic to where and is the stabiliser group of any partition of [d] into disjoint sets of sizes Clearly is isomorphic to the product so we have:
Proposition. The product corresponds to:
where acts on via
First use the following result, whose proof is an easy exercise if you know tensor products (here, tensor products without subscripts are over C).
- Suppose are -algebras, are subalgebras, and are -modules respectively. Then
In particular, take
for subgroups and . Since as -algebras, we have:
This gives So we have:
corresponding to as desired. ♦
Thus we obtain a graded ring isomorphism between and the set of virtual representations of all up to isomorphism. And we have:
Consider the case where W is trivial and e=1, so the product operation gives
On the other hand the trivial representation for corresponds to By Pieri’s formula, where runs through all partitions obtained from by adding 1 box. Hence:
By Frobenius reciprocity, we have, for any and ,
Hence, where the sum is over all partitions obtained from by removing one box.
Express and as a direct sums of irreps where