Covering $Irrep(S_n)$ With Tensor Products and Powers

Mark Sellke

Published 2020-04-11Version 1

We study when a tensor product of irreducible representations of the symmetric group $S_n$ contains all irreducibles as subrepresentations - we say such a tensor product covers $Irrep(S_n)$. Our results show that this behavior is typical. We first give a general criterion for such a tensor product to have this property. Using this criterion we show that the tensor product of a constant number of random irreducibles covers $Irrep(S_n)$ asymptotically almost surely. We also consider, for a fixed irreducible representation, the degree of tensor power needed to cover $Irrep(S_n)$. We show that the trivial lower bound based on dimension is tight up to a universal constant factor for every irreducible representation.