Concrete billiard arrays of polynomial type and Leonard systems

Abstract

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