Hall-Littlewood plane partitions and KP

TL;DR

This paper extends the generating function of plane partitions to Hall-Littlewood polynomials and shows its connection to KP tau-functions using fermionic methods and finite base restrictions.

Contribution

It provides a Fock space derivation of the Hall-Littlewood extension and links finite plane partition generating functions to KP tau-functions.

Findings

Derived a Fock space representation of the Hall-Littlewood generating function.

Established that finite s-by-s plane partition generating functions are KP tau-functions.

Connected Hall-Littlewood polynomial-based generating functions to integrable systems.

Abstract

MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletic to a generating function of weighted plane partitions that is related to Hall-Littlewood polynomials, S(t), and further to one related to Macdonald polynomials, S(t,q). Using Jing's 1-parameter deformation of charged free fermions, we obtain a Fock space derivation of the Hall-Littlewood extension. Confining the plane partitions to a finite s-by-s square base, we show that the resulting generating function, S_{s-by-s}(t), is an evaluation of a tau-function of KP.