Kronecker Coefficients, Crystals, and Bitableaux

TL;DR

This paper introduces bitableaux as a new combinatorial framework to interpret Kronecker coefficients, providing partial progress towards a combinatorial understanding and connecting it to generalized insertion algorithms.

Contribution

It proposes a new class of combinatorial objects called bitableaux and formulates a problem that could lead to a combinatorial interpretation of Kronecker coefficients.

Findings

Derived a combinatorial expansion for Kronecker product of Schur functions in the monomial basis.

Linked the problem of interpreting Kronecker coefficients to generalizations of RSK algorithms.

Made partial progress towards a combinatorial interpretation of Kronecker coefficients.

Abstract

What might a combinatorial interpretation of the Kronecker coefficients even look like? We introduce a class of combinatorial objects called bitableaux, which we believe are a natural candidate, and we formulate a purely combinatorial problem which if resolved would give a combinatorial interpretation of the Kronecker coefficients. We make some partial progress on this problem -- enough to extract a combinatorial expansion for a Kronecker product of Schur functions in the monomial basis. We also explain how in this framework finding a combinatorial interpretation for Kronecker coefficients can be thought of as looking for a generalization of the RSK and dual RSK insertion algorithms.