Constructing Hall-Littlewood Functions via a Deformation of the Bernstein Operator

TL;DR

This paper introduces a $t$-analogue of the Bernstein operator to construct Hall-Littlewood functions, revealing new combinatorial structures and proving stability of structure coefficients like Hall polynomials.

Contribution

It develops a $t$-deformation of the Bernstein operator enabling explicit construction of Jing's vertex operator for Hall-Littlewood functions.

Findings

Constructed a $t$-analogue of the Bernstein operator.

Provided an explicit construction of Jing's vertex operator.

Proved stability of certain structure coefficients, including Hall polynomials.

Abstract

The Bernstein operator $\mathbf{B}_n$ acts on a Schur function $S_\lambda$ by appending a part to the index, i.e., $\mathbf{B}_n S_\lambda=S_{(n,\lambda)}$. This provides a method of constructing the vertex operator representation of Schur functions since its homogeneous components are essentially just these Bernstein operators. Meanwhile, the Hall-Littlewood functions are an important generalization of the Schur functions, and they also have a vertex operator representation due to Jing. In this paper, we construct a $t$-analogue of the Bernstein operator, which allows for an explicit construction of the Jing operator. We show that the usual involution $\omega$ is fundamental to this construction, revealing further combinatorial structure. As an application, we use this vertex operator to prove stability of certain structure coefficients, including the Hall polynomials.