Sums of squares of the Littlewood-Richardson coefficients and GL(n)-harmonic polynomials

TL;DR

This paper explores the relationship between Littlewood-Richardson coefficients, Hilbert series, and GL(n)-harmonic polynomials within invariant theory, revealing new connections through symmetric functions and specializations.

Contribution

It introduces a stable symmetric function describing the Hilbert series of invariant rings and links it to sums of squares of Littlewood-Richardson coefficients and GL(n)-harmonic polynomials.

Findings

Hilbert series expressed as sums of squares of Littlewood-Richardson coefficients

Principal specialization relates to invariant subring Hilbert series

Other specializations yield additional insights into invariant theory

Abstract

We consider the example from invariant theory concerning the conjugation action of the general linear group on several copies of the $n \times n$ matrices, and examine a symmetric function which stably describes the Hilbert series for the invariant ring with respect to the multigradation by degree. The terms of this Hilbert series may be described as a sum of squares of Littlewood-Richardson coefficients. A "principal specialization" of the gradation is then related to the Hilbert series of the $\K$-invariant subring in the $\GL_n$-harmonic polynomials, where $\K$ denotes a block diagonal embedding of a product of general linear groups. We also consider other specializations of this Hilbert series.