A Basis for the GL_n Tensor Product Algebra

TL;DR

This paper provides an explicit basis for the $GL_n$ tensor product algebra, connecting representation theory, combinatorics, and invariant theory, and offering a new perspective on Littlewood-Richardson coefficients.

Contribution

It introduces a new explicit basis for the $GL_n$ tensor product algebra, linking combinatorial Littlewood-Richardson tableaux with classical invariant theory.

Findings

Explicit basis for the $GL_n$ tensor product algebra

Connection between basis and Littlewood-Richardson tableaux

Recasting of Littlewood-Richardson rule in invariant theory context

Abstract

This paper focuses on the $GL_n$ tensor product algebra, which encapsulates the decomposition of tensor products of arbitrary finite dimensional irreducible representations of $GL_n$. We will describe an explicit basis for this algebra. This construction relates directly with the combinatorial description of Littlewood-Richardson coefficients in terms of Littlewood-Richardson tableaux. Philosophically, one may view this construction as a recasting of the Littlewood-Richardson rule in the context of classical invariant theory.