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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.08422 |
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| _version_ | 1866918273566638080 |
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| author | Gutiérrez, Álvaro Martínez, Álvaro L. Szwej, Michał Wildon, Mark |
| author_facet | Gutiérrez, Álvaro Martínez, Álvaro L. Szwej, Michał Wildon, Mark |
| contents | We present a combinatorial proof of the $q$-Pfaff--Saalschütz identity by a composition of explicit bijections, in which $q$-binomial coefficients are interpreted as counting subspaces of $\mathbb{F}_q$-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig's integral form $\mathcal{U}_{\mathbb{Z}[q, q^{-1}]}(\mathfrak{sl}_2)$ of the Cartan subalgebra of the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_08422 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A new bijective proof of the $q$-Pfaff--Saalschütz identity with applications to quantum groups Gutiérrez, Álvaro Martínez, Álvaro L. Szwej, Michał Wildon, Mark Combinatorics Quantum Algebra 05A30 (Primary) 16T20, 05A19 (Secondary) We present a combinatorial proof of the $q$-Pfaff--Saalschütz identity by a composition of explicit bijections, in which $q$-binomial coefficients are interpreted as counting subspaces of $\mathbb{F}_q$-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig's integral form $\mathcal{U}_{\mathbb{Z}[q, q^{-1}]}(\mathfrak{sl}_2)$ of the Cartan subalgebra of the quantum group $\mathcal{U}_q(\mathfrak{sl}_2)$. |
| title | A new bijective proof of the $q$-Pfaff--Saalschütz identity with applications to quantum groups |
| topic | Combinatorics Quantum Algebra 05A30 (Primary) 16T20, 05A19 (Secondary) |
| url | https://arxiv.org/abs/2505.08422 |