Conjectures on the reduced Kronecker coefficients

TL;DR

This paper proposes conjectures on stable tensor products of symmetric group representations, related to reduced Kronecker coefficients, generalizing known log-concavity conjectures, with some cases proven and implications discussed.

Contribution

It introduces new conjectures on stable tensor products and reduced Kronecker coefficients, extending existing log-concavity results, with partial proofs and discussions.

Findings

Conjectures formulated on stable tensor products and reduced Kronecker coefficients.

Partial proofs provided for special cases of the conjectures.

Discussion of implications and connections to existing log-concavity theorems.

Abstract

We formulate a series of conjectures on the stable tensor product of irreducible representations of symmetric groups, which are closely related to the reduced Kronecker coefficients. These conjectures are certain generalizations of Okounkov's conjecture on the log-concavity of the Littlewood--Richardson coefficients and the Schur log-concavity theorem of Lam--Postnikov--Pylyavskyy. We prove our conjectures in some special cases and discuss some implications of these conjectures.