A decomposition rule for certain tensor product representations of the symmetric groups

TL;DR

This paper introduces a combinatorial rule for decomposing the tensor product of two irreducible symmetric group representations, specifically when one is a hook-shaped representation, simplifying calculations in representation theory.

Contribution

It provides a new combinatorial method to compute Kronecker products involving hook-shaped representations of symmetric groups.

Findings

Derived a explicit combinatorial rule for specific tensor products

Simplified the calculation of Kronecker products with hook representations

Enhanced understanding of the structure of symmetric group representations

Abstract

In this paper, we give a combinatorial rule to calculate the decomposition of the tensor product (Kronecker product) of two irreducible complex representations of the symmetric group ${\mathfrak S}_n$, when one of the representations corresponds to a hook $(n-m, 1^m)$.