A bilinear form relating two Leonard systems

TL;DR

This paper introduces and characterizes a balanced bilinear form relating two Leonard systems, which is significant in the study of $Q$-polynomial distance-regular graphs, providing new insights into their algebraic structure.

Contribution

It defines and analyzes a balanced bilinear form between Leonard systems, offering a novel perspective in the algebraic study of distance-regular graphs.

Findings

Characterization of the balanced bilinear form from multiple viewpoints

Connection established between Leonard systems and $Q$-polynomial distance-regular graphs

Enhanced understanding of the algebraic structure underlying Leonard systems

Abstract

Let $\Phi$, $\Phi'$ be Leonard systems over a field $\mathbb{K}$, and $V$, $V'$ the vector spaces underlying $\Phi$, $\Phi'$, respectively. In this paper, we introduce and discuss a balanced bilinear form on $V\times V'$. Such a form naturally arises in the study of $Q$-polynomial distance-regular graphs. We characterize a balanced bilinear form from several points of view.