A simplified Kronecker rule for one hook shape

TL;DR

This paper simplifies the combinatorial rule for calculating Kronecker coefficients when one partition is a hook shape by removing the need for conversion, making the rule easier to understand and prove.

Contribution

It provides a new characterization of colored Yamanouchi tableaux that simplifies the existing Kronecker rule for one hook shape, avoiding the conversion process.

Findings

Simplified the combinatorial rule for Kronecker coefficients with hook shape

Provided a new characterization of colored Yamanouchi tableaux

Streamlined the proof of the Kronecker rule for one hook shape

Abstract

Recently Blasiak gave a combinatorial rule for the Kronecker coefficient $g_{\lambda \mu \nu}$ when $\mu$ is a hook shape by defining a set of colored Yamanouchi tableaux with cardinality $g_{\lambda\mu\nu}$ in terms of a process called conversion. We give a characterization of colored Yamanouchi tableaux that does not rely on conversion, which leads to a simpler formulation and proof of the Kronecker rule for one hook shape.