Bounds on Kronecker and Littlewood-Richardson coefficients

The Kronecker coefficients of the symmetric group are the fundamental structure constants in the expansion of tensor products of irreducible symmetric group representations into irreducibles. Ever since their introduction by Murnaghan 80 years ago, they have puzzled algebraists and combinatorialists as very little about their nature is still known. Notably, we don't have any positive formula for them, in other words we don't yet know whether they are even #P-complete to compute.

In this talk we will discuss how to use various symmetric function identities and other combinatorial relationships to extract information about their size, especially upper bounds in various asymptotic regimes. We will also discuss bounds on their simpler GL analogues, the Littlewood-Richardson coefficients. This is based on joint works with Igor Pak and Damir Yeliussizov.