$Q$-polynomial distance-regular graphs and a double affine Hecke algebra of rank one

TL;DR

This paper explores the connection between $Q$-polynomial distance-regular graphs of $q$-Racah type and the double affine Hecke algebra of rank one, establishing a module structure that links algebraic and combinatorial properties.

Contribution

It constructs an $ ilde{H}_q$-module structure on a vector space derived from a $Q$-polynomial distance-regular graph, connecting graph adjacency matrices with Hecke algebra elements.

Findings

The module structure relates adjacency matrices to Hecke algebra generators.

It characterizes dual adjacency matrices via algebraic actions.

Uses Leonard systems to establish the algebraic-combinatorial link.

Abstract

We study a relationship between $Q$-polynomial distance-regular graphs and the double affine Hecke algebra of type $(C^{\vee}_1,C_1)$. Let $\Gamma$ denote a $Q$-polynomial distance-regular graph with vertex set $X$. We assume that $\Gamma$ has $q$-Racah type and contains a Delsarte clique $C$. Fix a vertex $x \in C$. We partition $X$ according to the path-length distance to both $x$ and $C$. This is an equitable partition. For each cell in this partition, consider the corresponding characteristic vector. These characteristic vectors form a basis for a $\mathbb{C}$-vector space ${\bf W}$. The universal double affine Hecke algebra of type $(C^{\vee}_1,C_1)$ is the $\mathbb{C}$-algebra $\hat{H}_q$ defined by generators $\{t^{\pm1}_n\}^3_{n=0}$ and relations (i) $t_nt_n^{-1}=t_n^{-1}t_n=1$; (ii) $t_n+t_n^{-1}$ is central; (iii) $t_0t_1t_2t_3 = q^{-1/2}$. In this paper, we display an $\hat{H}_q$-module structure for ${\bf W}$. For this module and up to affine transformation, (i) $t_0t_1+(t_0t_1)^{-1}$ acts as the adjacency matrix of $\Gamma$; (ii) $t_3t_0+(t_3t_0)^{-1}$ acts as the dual adjacency matrix of $\Gamma$ with respect to $C$; (iii) $t_1t_2+(t_1t_2)^{-1}$ acts as the dual adjacency matrix of $\Gamma$ with respect to $x$. To obtain our results we use the theory of Leonard systems.