Stability of coefficients in the Kronecker product of a hook and a rectangle

TL;DR

This paper proves a stability property for Kronecker coefficients involving hook and rectangle partitions, providing bounds for stability and methods to recover decompositions from minimal stable cases.

Contribution

It establishes a stability result for specific Kronecker coefficients and offers bounds and techniques to determine the stable decomposition.

Findings

Proves stability of Kronecker coefficients for hook and rectangle partitions.

Provides explicit bounds for the size of partitions where stability begins.

Shows that stable decompositions can be recovered from minimal stable cases.

Abstract

We use recent work of Jonah Blasiak (2012) to prove a stability result for the coefficients in the Kronecker product of two Schur functions: one indexed by a hook partition and one indexed by a rectangle partition. We also give bounds for the size of the partition starting with which the Kronecker coefficients are stable. Moreover, we show that once the bound is reached, no new Schur functions appear in the decomposition of Kronecker product, thus allowing one to recover the decomposition from the smallest case in which the stability holds.