On palindromic numerators of bigraded symmetric orbifold Hilbert series and Kostka-Foulkes polynomials

TL;DR

This paper reveals that palindromic numerators in bigraded symmetric orbifold Hilbert series can be expressed using Kostka-Foulkes polynomials, connecting them to KP $ au$-functions and differential operators.

Contribution

It introduces a novel link between palindromic numerators, Kostka-Foulkes polynomials, and KP hierarchy, providing new insights into the structure of Hilbert series in this context.

Findings

Palindromic numerators are sums of Kostka-Foulkes polynomials.

These polynomials are eigenvalues of a differential operator.

The work connects Hilbert series with KP $ au$-functions.

Abstract

From our work on partition functions in log gravity, we show that the palindromic numerators in two variables of bigraded symmetric orbifold Hilbert series take the form of sums of products of Kostka-Foulkes polynomials associated with a pair of partition $\lambda$ and $\mu=(1^n)$. The log partition function also being a KP $\tau$-function, our work gives a new description of Hall-Littlewood and Kostka-Foulkes polynomials as palindromic numerators of quotient expansions in the moduli space of formal power series solutions of the KP hierarchy. Using the structure and properties of the log partition function, we also show that the palindromic polynomials are eigenvalues of a differential operator arising from a recurrence relation and acting on the Hilbert series.