A new bijective proof of the $q$-Pfaff--Saalschütz identity with applications to quantum groups

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 ${\mathcal{U}}_{\mathbb{Z}[q,q^{-1}]}({\mathfrak{s}\mathfrak{l}}_2)$ of the Cartan subalgebra of the quantum group ${\mathcal{U}}_q({\mathfrak{s}\mathfrak{l}}_2)$.