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

TL;DR

The paper tackles the positivity of Kronecker coefficients in the context of Saxl's conjecture, which posits that the Kronecker square $[\rho_k]^2$ contains all irreducible characters of the symmetric group $\mathfrak{S}_n$ with $n=\binom{k+1}{2}$. By leveraging the semi-group property of Kronecker coefficients and generalized $t$-blocks, it constructs families of nonzero constituents, notably through the telescope framework and the sets $\mathrm{Kron}(\rho_m)$, showing that a wide class of partitions (including telescopic ones) belong to $\mathrm{Kron}(\rho_k)$. The core result demonstrates that for suitable parameters and constructions, $g(\rho_k,\rho_k,\alpha)\neq 0$ for many $\alpha$, yielding partial progress towards Saxl's conjecture and yielding corollaries about hook and block-structured constituents. Overall, the work provides a principled, combinatorial method to generate explicit constituents in $[\rho_k]^2$, highlighting the utility of semi-group techniques and generalized blocks in understanding Kronecker positivity.

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.

Theorems & Definitions (18)

• Proposition 1.1
• Corollary 1.2
• Definition 1.3
• Proposition 1.4
• Definition 1.5
• Theorem 1.6
• Corollary 1.7
• Lemma 2.1
• Lemma 2.2
• proof