Concrete Billiard Arrays of Polynomial Type and Leonard Systems

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 $0\le i\le d$, the $i^{\mathrm{t}\mathrm{h}}$ 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 $\mathrm{\Phi }=(A;\{E_i{\}}_{i=0}^d;A^{\ast };\{E_i^{\ast }{\}}_{i=0}^d)$ 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 $\mathcal{L}$ corresponds to the $\mathrm{\Phi }$-split (resp. ${\mathrm{\Phi }}^{\Downarrow }$-split) decomposition of $V$.