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
Proof
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
Proof
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. ♦
Consequences
Ring Isomorphism
Thus we obtain a graded ring isomorphism between and the set of virtual representations of all
up to isomorphism. And we have:
Pieri’s Formulae
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:
For example,
Restriction
By Frobenius reciprocity, we have, for any and
,
Hence, where the sum is over all partitions
obtained from
by removing one box.
Exercise
Express and
as a direct sums of irreps where