Path operators and $(q,t)$-tau functions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916782408728576 |
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| author | Dali, Houcine Ben Bonzom, Valentin Dołęga, Maciej |
| author_facet | Dali, Houcine Ben Bonzom, Valentin Dołęga, Maciej |
| contents | We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up and down. We demonstrate that positive linear combinations of these operators are the images of Negut elements via a representation of the shuffle algebra acting on the space of symmetric functions. Additionally, we provide a monomial, elementary, and Schur symmetric function expansion for the symmetric function obtained through repeated applications of the path operators on $1$.
We apply path operators to investigate a $(q,t)$-deformation of the classical hypergeometric tau functions, which generalizes several important series already present in enumerative geometry, gauge theory, and integrability. We prove that this function is uniquely characterized by a family of partial differential equations derived from a positive linear combination of path operators. We also use our operators to offer a new, independent proof of the key result in establishing the extended delta conjecture of Haglund, Remmel, and Wilson. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_06036 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Path operators and $(q,t)$-tau functions Dali, Houcine Ben Bonzom, Valentin Dołęga, Maciej Combinatorics Mathematical Physics Quantum Algebra Representation Theory We construct a new class of operators that act on symmetric functions with two deformation parameters $q$ and $t$. Our combinatorial construction associates each operator with a specific lattice path, whose steps alternate between moving up and down. We demonstrate that positive linear combinations of these operators are the images of Negut elements via a representation of the shuffle algebra acting on the space of symmetric functions. Additionally, we provide a monomial, elementary, and Schur symmetric function expansion for the symmetric function obtained through repeated applications of the path operators on $1$. We apply path operators to investigate a $(q,t)$-deformation of the classical hypergeometric tau functions, which generalizes several important series already present in enumerative geometry, gauge theory, and integrability. We prove that this function is uniquely characterized by a family of partial differential equations derived from a positive linear combination of path operators. We also use our operators to offer a new, independent proof of the key result in establishing the extended delta conjecture of Haglund, Remmel, and Wilson. |
| title | Path operators and $(q,t)$-tau functions |
| topic | Combinatorics Mathematical Physics Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2506.06036 |