The problem of deciding the positivity of Kronecker coefficients and Saxl conjecture

TL;DR

This paper explores the positivity of Kronecker coefficients and the Saxl conjecture, focusing on methods to identify irreducible representations in tensor squares of symmetric group representations.

Contribution

It introduces and analyzes two methods—the semi-group property and generalized blocks—to determine irreducible components related to the Saxl conjecture.

Findings

Semi-group property aids in identifying positive Kronecker coefficients.

Generalized blocks provide a framework for decomposing tensor products.

Results contribute to understanding the Saxl conjecture's validity.

Abstract

Given an positive integer $k$, let $n:=\binom{k+1}{2}$. In 2012, during a talk at UCLA, Jan Saxl conjectured that all irreducible representations of the symmetric group $S_n$ occur in the decomposition of the tensor square of the irreducible representation corresponding to the staircase partition. In this paper, we investigate two useful methods to obtain some irreducible representations that occur in this decomposition. Our main tolls are the semi-group property for Kronecker coefficients and generalized blocks of symmetric groups.