Introduction
Let's fix . For each n, the unitary group is represented in the space of tensors of rank over
and the symmetric group acts on by permutation of factors. Now the space breaks into the direct sum of subspaces which are primary with respect to each of these actions and irreducible with respect to the joint action of
Where ranges over all Young diagrams of size with at most rows (So means is a partition of with ). The tensors from are said to have symmetry type . We define the relative dimensions
which tell us how tensors are distributed into symmetry types.
Motivation and Question
I was reading Kerov's "Asymptotic representation theory of the symmetric group and it's aplications in analysis" and was trying to provide proofs for some of the results stated there. (He does give references, which I can't reach at the moment.)
The following two theorems are due to Kerov
Theorem 1 If for each we associate with . The joint distribution of 's as with respect to the measure on weakly converges to an absolutely continuous measure on the cone
with density
where
.
Theorem 2 Let be the Young diagram for which the tensors of type are most probable. Then
for each where are the roots of the Hermite polynomial
I can prove Theorem 2 assuming Theorem 1, but I don't see a nice argument for proving the first theorem itself. Can anyone provide a sketch of the proof, or some hint how to approach Theorem 1?
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