Dual Polar Graphs, a nil-DAHA of Rank One, and Non-Symmetric Dual q-Krawtchouk Polynomials

TL;DR

This paper constructs a module for a nil-DAHA from dual polar graphs and uses it to define and analyze non-symmetric dual q-Krawtchouk polynomials, revealing their recurrence and orthogonality relations.

Contribution

It introduces a new connection between dual polar graphs, nil-DAHA modules, and non-symmetric dual q-Krawtchouk polynomials, expanding the algebraic combinatorics framework.

Findings

Constructs a 2D-dimensional irreducible module for nil-DAHA from dual polar graphs.

Defines non-symmetric dual q-Krawtchouk polynomials using the module.

Derives recurrence and orthogonality relations for these polynomials.

Abstract

Let $\Gamma$ be a dual polar graph with diameter $D \geqslant 3$, having as vertices the maximal isotropic subspaces of a finite-dimensional vector space over the finite field $\mathbb{F}_q$ equipped with a non-degenerate form (alternating, quadratic, or Hermitian) with Witt index $D$. From a pair of a vertex $x$ of $\Gamma$ and a maximal clique $C$ containing $x$, we construct a $2D$-dimensional irreducible module for a nil-DAHA of type $(C^{\vee}_1, C_1)$, and establish its connection to the generalized Terwilliger algebra with respect to $x$, $C$. Using this module, we then define the non-symmetric dual $q$-Krawtchouk polynomials and derive their recurrence and orthogonality relations from the combinatorial points of view. We note that our results do not depend essentially on the particular choice of the pair $x$, $C$, and that all the formulas are described in terms of $q$, $D$, and one other scalar which we assign to $\Gamma$ based on the type of the form.