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

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.