Kronecker coefficients and noncommutative super Schur functions

TL;DR

This paper introduces noncommutative super Schur functions to derive a positive combinatorial rule for Kronecker coefficients involving hook partitions, linking it to previous heuristics and expanding the combinatorial understanding of these coefficients.

Contribution

It develops a new theory of noncommutative super Schur functions and provides a positive combinatorial rule for specific Kronecker coefficients, connecting to earlier heuristics.

Findings

Established a positive combinatorial rule for Kronecker coefficients with hook partitions.

Connected the rule to Lascoux's heuristic for Kronecker coefficients.

Recovered previous results of the authors on Kronecker coefficients.

Abstract

The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene. We develop a theory of noncommutative super Schur functions and use it to prove a positive combinatorial rule for the Kronecker coefficients where one of the partitions is a hook, recovering previous results of the two authors. This method also gives a precise connection between this rule and a heuristic for Kronecker coefficients first investigated by Lascoux.