Generalized Q-functions for GKM

TL;DR

This paper explores a generalization of $Q$ Schur functions using Hall-Littlewood polynomials at roots of unity to expand the partition function of the generalized Kontsevich model (GKM), revealing new mathematical structures.

Contribution

It introduces a new class of functions $Q^{(n)}$ based on Hall-Littlewood polynomials for arbitrary $n>2$, extending the $Q$ Schur functions to GKM with monomial potentials.

Findings

$Q^{(n)}$ functions are associated with $n$-strict Young diagrams

Partition function of GKM can be expanded in $Q^{(n)}$ functions

Coefficients of expansion are not yet fully understood or characterized

Abstract

Recently we explained that the classical $Q$ Schur functions stand behind various well-known properties of the cubic Kontsevich model, and the next step is to ask what happens in this approach to the generalized Kontsevich model (GKM) with monomial potential $X^{n+1}$. We propose to use the Hall-Littlewood polynomials at the parameter equal to the $n$-th root of unity as a generalization of the $Q$ Schur functions from $n=2$ to arbitrary $n>2$. They are associated with $n$-strict Young diagrams and are independent of time-variables $p_{kn}$ with numbers divisible by $n$. These are exactly the properties possessed by the generalized Kontsevich model (GKM), thus its partition function can be expanded in such functions $Q^{(n)}$. However, the coefficients of this expansion remain to be properly identified. At this moment, we have not found any "superintegrability" property $<character>\,\sim character$, which expressed these coefficients through the values of $Q$ at delta-loci in the $n=2$ case. This is not a big surprise, because for $n>2$ our suggested $Q$ functions are not looking associated with characters.