Splitting Kronecker squares, 2-decomposition numbers, Catalan Combinatorics, and the Saxl conjecture

TL;DR

This paper explores the splitting of Kronecker squares of symmetric group characters, introduces new results and conjectures, and connects these to 2-modular decomposition, Catalan combinatorics, and the Saxl conjecture to deepen understanding.

Contribution

It provides new results and conjectures on the splitting of Kronecker squares and links these to combinatorial and modular representation theory topics.

Findings

New results on Kronecker square splitting

Connections to Catalan combinatorics and Saxl conjecture

Discussion of implications for 2-modular decomposition

Abstract

While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and conjectures) are presented on this splitting of the squares that contribute to a refined understanding of the Kronecker squares. Furthermore, connections to 2-modular decomposition numbers, Catalan combinatorics, and to the Saxl conjecture are discussed which further motivate the study of these splittings.