Geometric invariant theory and stretched Kostka quasi-polynomials

TL;DR

This paper uses geometric invariant theory to analyze the degrees and periods of stretched Kostka quasi-polynomials, resolving a conjecture and providing new geometric insights into their structure.

Contribution

It introduces a GIT-based method to determine the degree of stretched Kostka quasi-polynomials and resolves a conjecture on their explicit formula.

Findings

Degree of the quasi-polynomial equals the dimension of a GIT quotient

Resolved Gao and Gao's conjecture on the explicit degree formula

Provided computational evidence for the minimal period of the quasi-polynomial

Abstract

For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $\lambda, \mu$, we consider the dimensions of weight spaces $V_\lambda(\mu)$ of weight $\mu$ in the irreducible, finite-dimensional highest weight $\lambda$ representation. For natural numbers $N$, the function $N \mapsto \dim V_{N\lambda}(N\mu)$ is a quasi-polynomial in $N$, the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we realize the degree of this quasi-polynomial as the dimension of a certain GIT quotient. As a result, we resolve a conjecture of Gao and Gao on an explicit formula for this degree. We also discuss periods of this quasi-polynomial determined by the GIT approach, and give computational evidence supporting a geometric determination of the minimal period.