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arXiv · 2511.01114

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

John Graf
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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.

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John Graf. 2025-11-02. Constructing Hall-Littlewood Functions via a Deformation of the Bernstein Operator. https://arxiv.org/abs/2511.01114

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