Kronecker products and the RSK correspondence

TL;DR

This paper introduces a combinatorial bijection linking minimal matrices with prescribed marginals to Kronecker coefficients, generalizing the RSK correspondence to 3D matrices and providing new algorithms and descriptions.

Contribution

It presents a novel bijection that combinatorially realizes an identity relating minimal matrices and Kronecker coefficients, extending the RSK correspondence to 3D matrices.

Findings

Provides a bijection between minimal matrices and Kronecker components

Develops an algorithm associating minimal matrices to Kronecker coefficients

Generalizes the RSK correspondence to 3-dimensional matrices

Abstract

The starting point for this work is an identity that relates the number of minimal matrices with prescribed 1-marginals and coefficient sequence to a linear combination of Kronecker coefficients. In this paper we provide a bijection that realizes combinatorially this identity. As a consequence we obtain an algorithm that to each minimal matrix associates a minimal component, with respect to the dominance order, in a Kronecker product, and a combinatorial description of the corresponding Kronecker coefficient in terms of minimal matrices and tableau insertion. Our bijection follows from a generalization of the dual RSK correspondence to 3-dimensional binary matrices, which we state and prove. With the same tools we also obtain a generalization of the RSK correspondence to 3-dimensional integer matrices.