Highest Weight Generating Functions for Hilbert Series

TL;DR

This work develops a highest-weight generating function (HWG) framework to encode and analyze refined Hilbert series via Dynkin-label data, leveraging Plethystic Exponential/Log and Weyl integration. The approach yields finite, rational generating functions $g^{G}(m_i,\mathcal{X})$ and enables extraction of representation data for moduli spaces of SQCD theories and instanton configurations, including exceptional groups like $G_2$. By applying HWGs to gauge-invariant operators and instanton moduli spaces, the paper reveals palindromic, Calabi–Yau–like structures and clarifies when Hilbert series decompose into sums over flavor irreps, offering scalable methods for high-rank groups. The results connect invariant tensor structures, tensor-product decompositions, and F-term constraints to concrete HWG descriptions, providing a compact, constructive language for the combinatorics of group representations in product-group theories. The framework paves the way for analyzing a wide class of moduli spaces and could inform geometric and algebraic properties of these spaces beyond the specific SQCD and instanton examples studied.

Abstract

We develop a new method for representing Hilbert series based on the highest weight Dynkin labels of their underlying symmetry groups. The method draws on plethystic functions and character generating functions along with Weyl integration. We give explicit examples showing how the use of such highest weight generating functions (HWGs) permits an efficient encoding and analysis of the Hilbert series of the vacuum moduli spaces of classical and exceptional SQCD theories and also of the moduli spaces of instantons. We identify how the HWGs of gauge invariant operators of a selection of classical and exceptional SQCD theories result from the interaction under symmetrisation between a product group and the invariant tensors of its gauge group. In order to calculate HWGs, we derive and tabulate character generating functions for low rank groups by a variety of methods, including a general character generating function, based on the Weyl Character Formula, for any classical or exceptional group.