Kronecker coefficients via the Giambelli identity for Schur functions

TL;DR

This paper develops a new framework combining classical identities and existing rules to interpret and evaluate Kronecker coefficients, extending combinatorial rules to broader cases.

Contribution

It introduces a systematic approach that reduces Kronecker coefficient analysis to hook-indexed cases, enabling broader combinatorial interpretations.

Findings

Provides combinatorial interpretations for specific Kronecker coefficients involving two-row and hook-like partitions.

Reduces the study of Kronecker coefficients to alternating sums involving simpler cases.

Extends hook-based combinatorial rules to wider families of Kronecker coefficients.

Abstract

One of the central open problems in both algebraic combinatorics and representation theory is to find a positive combinatorial rule for Kronecker coefficients $ g_{\lambda \, \mu \, \nu}$. A notable advance in this direction is due to Blasiak, who proved a combinatorial interpretation in terms of colored Yamanouchi tableaux for the case whereby one of the indexing partitions is hook-shaped. In this paper, we introduce a framework for the evaluation and combinatorial interpretation of Kronecker coefficients, combining a Schur function identity of Littlewood, the Giambelli identity for Schur functions, and Blasiak's combinatorial rule. This framework reduces the study of Kronecker coefficients to alternating sums involving hook-indexed cases. As an application of this framework, we obtain combinatorial interpretations of $g_{t, h^{(1)}, h^{(2)}}$ for two-row partitions $t$ and hook-like partitions $h^{(1)}$ and $h^{(2)}$ satisfying natural conditions. More broadly, our approach provides a systematic method for extending hook-based combinatorial rules to wider families of Kronecker coefficients.