Grassmann graphs, degenerate DAHA, and non-symmetric dual $q$-Hahn polynomials

Abstract

We discuss the Grassmann graph $J_q(N,D)$ with $N\ge 2D$, having as vertices the $D$-dimensional subspaces of an $N$-dimensional vector space over the finite field ${\mathbb{F}}_q$. This graph is distance-regular with diameter $D$; to avoid trivialities we assume $D\ge 3$. Fix a pair of a Delsarte clique $C$ of $J_q(N,D)$ and a vertex $x$ in $C$. We construct a $2D$-dimensional irreducible module $\mathbf{W}$ for the Terwilliger algebra $\mathbf{T}$ of $J_q(N,D)$ associated with the pair $x$, $C$. We show that $\mathbf{W}$ is an irreducible module for the confluent Cherednik algebra ${\mathcal{H}}_{\mathrm{V}}$ and describe how the $\mathbf{T}$-action on $\mathbf{W}$ is related to the ${\mathcal{H}}_{\mathrm{V}}$-action on $\mathbf{W}$. Using the ${\mathcal{H}}_{\mathrm{V}}$-module $\mathbf{W}$, we define non-symmetric dual $q$-Hahn polynomials and prove their recurrence and orthogonality relations from a combinatorial viewpoint.

31 pages, 3 figures