On the Kronecker product of Schur functions of square shapes

TL;DR

This paper investigates the tensor squares of irreducible representations of symmetric groups associated with square Young diagrams, providing a formula for Kronecker coefficients and proving positivity in specific cases.

Contribution

It introduces a new formula for Kronecker coefficients involving square partitions and establishes positivity results for certain families of partitions.

Findings

Derived a formula for Kronecker coefficients with square partitions

Proved positivity of square Kronecker coefficients for three-row partitions

Confirmed positivity for near-hook partitions

Abstract

Motivated by the Saxl conjecture and the tensor square conjecture, which states that the tensor squares of certain irreducible representations of the symmetric group contain all irreducible representations, we study the tensor squares of irreducible representations associated with square Young diagrams. We give a formula for computing Kronecker coefficients, which are indexed by two square partitions and a three-row partition, specifically one with a short second row and the smallest part equal to 1. We also prove the positivity of square Kronecker coefficients for particular families of partitions, including three-row partitions and near-hooks.