Linear Transformations of Vertex Operators of Hall–Littlewood Polynomials

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 BKP hierarchies. 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.