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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.01114 |
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| _version_ | 1866915592427012096 |
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| author | Graf, John |
| author_facet | Graf, John |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_01114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing Hall-Littlewood Functions via a Deformation of the Bernstein Operator Graf, John Combinatorics 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. |
| title | Constructing Hall-Littlewood Functions via a Deformation of the Bernstein Operator |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2511.01114 |