On tensor products of polynomial representations

TL;DR

This paper characterizes when tensor products of irreducible polynomial representations of GL(n,C) are isomorphic, revealing new families of non-zero Littlewood-Richardson coefficients and conditions for Schur non-negativity.

Contribution

It provides necessary and sufficient combinatorial conditions for tensor product isomorphisms of polynomial representations of GL(n,C).

Findings

Identifies new families of non-zero Littlewood-Richardson coefficients.

Establishes a condition on Schur non-negativity.

Provides a complete combinatorial characterization for tensor product isomorphisms.

Abstract

We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of $GL(n,\mathbb{C})$ is isomorphic to another. As a consequence we discover families of Littlewood-Richardson coefficients that are non-zero, and a condition on Schur non-negativity.