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

TL;DR

This paper provides a combinatorial bijective proof of the $q$-Pfaff--Saalsch"utz identity, leading to new insights into quantum binomial coefficients and a novel presentation of Lusztig's integral form of the quantum group $ ext{U}_q( ext{sl}_2)$.

Contribution

It introduces a new bijective proof of the $q$-Pfaff--Saalsch"utz identity and derives a new multiplication rule for quantum binomial coefficients, impacting quantum group theory.

Findings

New bijective proof of the $q$-Pfaff--Saalsch"utz identity

A new multiplication rule for quantum binomial coefficients

A new presentation of Lusztig's integral form of $ ext{U}_q( ext{sl}_2)$

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 $\mathcal{U}_q(\mathfrak{sl}_2)$.