Concrete Billiard Arrays of Polynomial Type and Leonard Systems

TL;DR

This paper constructs Concrete Billiard Arrays of polynomial type linked to Leonard systems, revealing their boundary decompositions correspond to specific split decompositions of the vector space.

Contribution

It introduces a new class of Billiard Arrays associated with polynomial eigenvalue structures and connects them to Leonard systems and their split decompositions.

Findings

Constructed Concrete Billiard Arrays with polynomial type.

Established correspondence between array boundaries and Leonard system split decompositions.

Provided normalization conditions linking arrays to Leonard systems.

Abstract

Let $d$ denote a nonnegative integer and let $\mathbb{F}$ denote a field. Let $V$ denote a $d+1$ dimensional vector space over $\mathbb{F}$. Given an ordering $\{\theta_i\}_{i=0}^d$ of the eigenvalues of a multiplicity-free linear map $A: V \to V$, we construct a Concrete Billiard Array $\mathcal{L}$ with the property that for $0 \leq i \leq d$, the $i^{\rm th}$ vector on its bottom border is in the $\theta_i$-eigenspace of $A$. The Concrete Billiard Array $\mathcal{L}$ is said to have polynomial type. We also show the following. Assume that there exists a Leonard system $\Phi=(A;\{E_i\}_{i=0}^d;A^*;\{E_i^*\}_{i=0}^d)$ where $E_i$ is the primitive idempotent of $A$ corresponding to $\theta_i$ for $0 \leq i \leq d$. Then, we show that after a suitable normalization, the left (resp. right) boundary of $\mathcal{L}$ corresponds to the $\Phi$-split (resp. $\Phi^{\Downarrow}$-split) decomposition of $V$.