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arXiv · 2505.08422

A new bijective proof of the $q$-Pfaff--Saalsch\"utz identity with applications to quantum groups

Álvaro Gutiérrez·Álvaro L. Martínez·Michał Szwej·Mark Wildon
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Abstract

We present a combinatorial proof of the $q$-Pfaff--Saalsch\"utz 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 $\mathcal{U}_q(\mathfrak{sl}_2)$.

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Álvaro Gutiérrez, Álvaro L. Martínez, Michał Szwej, Mark Wildon. 2025-05-13. A new bijective proof of the $q$-Pfaff--Saalsch\"utz identity with applications to quantum groups. https://doi.org/10.1016/j.ejc.2025.104321

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