Constructing Hall-Littlewood Functions via a Deformation of the Bernstein Operator
John Graf
TL;DR
The paper develops a $t$-analogue of the Bernstein operator to realize Hall-Littlewood functions via a Jing-like vertex operator, with the involution $\omega$ playing a central role in linking the bases $Q_\lambda$ and $B_\lambda$. It constructs the Hall-Littlewood vertex operator through $H(z)=\alpha_z\beta_{-1/z}^\perp$ and proves a $t$-analogue of Bernstein-type creation operators that append a row to the index, producing $Q_{(n,\lambda)}$ from $Q_\lambda$ and $B_{(n,\lambda)}$ from $B_\lambda$. The work also establishes stability results for skew coefficients and Hall polynomials, showing that certain structure constants stabilize as the first part of the index grows, thereby yielding asymptotic stability for Hall polynomials. These contributions provide a combinatorial and operator-theoretic framework for Hall-Littlewood functions and their structure coefficients, with potential extensions to broader families of symmetric functions.
Abstract
The Bernstein operator $\mathbf{B}_n$ acts on a Schur function $S_λ$ by appending a part to the index, i.e., $\mathbf{B}_n S_λ=S_{(n,λ)}$. 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 $ω$ 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.