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

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.

14 pages, minor changes to match published version

References in corpus (4)

• Graviton 1-loop partition function for 3-dimensional massive gravity
• Integrable hierarchies, Hurwitz numbers and a branch point field in critical topologically massive gravity
• Fragmented perspective of self-organized criticality and disorder in log gravity
• Urn models, Markov chains and random walks in cosmological topologically massive gravity at the critical point

Cited by in corpus (1)

• Monodromy, Logarithmic Sectors, and Two-Point Functions in Critical Topologically Massive Gravity