Linear transformations of vertex operator presentations of Hall-Littlewood polynomials

TL;DR

This paper investigates how linear transformations affect quantum fields and vertex operator presentations of Hall-Littlewood polynomials, revealing their impact on symmetric functions and integrable hierarchies.

Contribution

It introduces a framework for understanding linear transformations of quantum fields and their effects on Hall-Littlewood polynomials, connecting to various symmetric functions and integrable systems.

Findings

Transformations preserve commutation relations.

Explicit combinatorial formulas derived.

Specializations describe all polynomial tau functions of KP and BKP hierarchies.

Abstract

We study the effect of linear transformations on quantum fields with applications to vertex operator presentations of symmetric functions. Properties of linearly transformed quantum fields and corresponding transformations of Hall-Littlewood polynomials are described, including preservation of commutation relations, stability, explicit combinatorial formulas and generating functions. We prove that specializations of linearly transformed Hall-Littlewood polynomials describe all polynomial tau functions of the KP and the BKP hierarchy. Examples of linear transformations are related to multiparameter symmetric functions, Grothendieck polynomials, deformations by cyclotomic polynomials, and some other variations of Schur symmetric functions that exist in the literature.