Geometric invariant theory and stretched Kostka quasi-polynomials

Marc Besson, Sam Jeralds and Joshua Kiers

Abstract

For \(G\) a simple, 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 quasi-polynomial in \(N\), the stretched Kostka quasi-polynomial. Using methods of geometric invariant theory (GIT), we compute the degree of this quasi-polynomial, resolving a conjecture of Gao and Gao. 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.

AMS Subject Classification: Primary 22E46; secondary 17B10, 14L24, 14M15.