Geometric invariant theory and stretched Kostka quasi-polynomials
Marc Besson, Sam Jeralds, Joshua Kiers
TL;DR
This work connects stretched Kostka quasi-polynomials $K_{\lambda,\mu}(N) = \dim V_{N\lambda}(N\mu)$ to geometric invariant theory by realizing the degree of these quasi-polynomials as the dimension of a GIT quotient $(G/P_\lambda) //_{\mathbb{L}} T$ with $\mathbb{L} = L_\lambda \otimes \mathbb{C}_\mu$. It provides a uniform proof that extends prior results (Gao–Gao) to all semisimple groups and confirms a conjectured degree formula, namely $\deg K_{\lambda,\mu}(N) = \tfrac{1}{2}|\Phi| - \tfrac{1}{2}|\Phi'| - \mathrm{rank}(\Phi)$ for primitive pairs, with $\Phi' = \mathrm{span}\{\alpha_i : d_i = 0\}$ when $\lambda = \sum_i d_i \varpi_i$. The paper also explains how the periods of these quasi-polynomials arise from descent data: the minimal $d$ such that $\mathbb{L}^{\otimes d}$ descends to the quotient gives a period, and it establishes descent criteria and corollaries, including the $\,\mu=0\,$ case recovering Kirillov–Reshetikhin results in type A and pertinently bounding periods in classical types. Finally, it provides concrete computational examples for types $G_2$, $B_3$, $D_4$, and $F_4$ to illustrate how the descent lattice governs the observed periods and degrees.
Abstract
For $G$ a semisimple, simply-connected complex algebraic group and two dominant integral weights $λ, μ$, we consider the dimensions of weight spaces $V_λ(μ)$ of weight $μ$ in the irreducible, finite-dimensional highest weight $λ$ representation. For natural numbers $N$, the function $N \mapsto \dim V_{Nλ}(Nμ)$ 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.