On the structures of hive algebras and tensor product algebras for general linear groups of low rank

TL;DR

This paper provides explicit algebraic descriptions of the tensor product algebra for low-rank general linear groups using hive models, offering new insights into representation decompositions and Littlewood-Richardson coefficients.

Contribution

It presents finite generators and relations for TA(n) for n=2, 3, 4, and computes generating functions for sums of Littlewood-Richardson coefficients, advancing understanding of these algebraic structures.

Findings

Finite presentations of TA(n) for n=2, 3, 4

Explicit description of highest weight vectors

Computed generating functions for Littlewood-Richardson sums

Abstract

The tensor product algebra TA(n) for the complex general linear group GL(n), introduced by Howe et al., describes the decomposition of tensor products of irreducible polynomial representations of GL(n). Using the hive model for the Littlewood-Richardson coefficients, we provide a finite presentation of the algebra TA(n) for n=2, 3, 4 in terms of generators and relations, thereby giving a description of highest weight vectors of irreducible representations in the tensor products. We also compute the generating function of certain sums of Littlewood-Richardson coefficients.