Kronecker products and the RSK correspondence

Abstract

A matrix M with nonnegative integer entries is minimal if the nonincreasing sequence of its entries (called π-sequence) is minimal, in the dominance order of partitions, among all nonincreasing sequences of entries of matrices with nonnegative integers that have the same 1-marginals as A.The starting point for this work is an identity that relates the number of minimal matrices that have fixed 1-marginals and π-sequence to a linear combination of Kronecker coefficients. In this paper we provide a bijection that realizes combinatorially this identity. From this bijection 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.