Concrete Billiard Arrays of Polynomial Type and 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 $V$.