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

TL;DR

The paper proves that the palindromic numerators in two variables of the bigraded symmetric orbifold Hilbert series can be expressed as sums of products of Kostka-Foulkes polynomials for partitions $\lambda$ and $\mu=(1^n)$. It situates Hall-Littlewood and Kostka-Foulkes polynomials within the framework of palindromic numerators arising from quotient expansions on an infinite-dimensional Grassmann manifold and identifies the log partition as a KP $\tau$-function. A key technical result is that these palindromic polynomials satisfy a differential-operator eigenvalue relation on the Hilbert series, derived via recurrences tied to cycle-index structures. The work connects symmetric-function theory, Macdonald/Hall-Littlewood polynomials, and algebraic geometry (Hilbert schemes) to a dynamicalKP-type picture for log gravity, suggesting new avenues for understanding AdS$_3$/LCFT$_2$ phenomenology and self-organized dynamics like KP hierarchies.

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 $λ$ and $μ=(1^n)$. The log partition function also being a KP $τ$-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.

Figures (1)

• Figure 1: Young diagram of a partition