Concrete Billiard Arrays of Polynomial Type and Leonard Systems
Jimmy Vineyard
TL;DR
This work connects Billiard Arrays to Leonard systems by constructing a Concrete Billiard Array of Polynomial Type from a multiplicity-free operator $A$ with eigenvalues $\{\theta_i\}$, ensuring the bottom border lies in the corresponding eigenspaces. When a Leonard system $\Phi$ exists with $E_i$ as the primitive idempotents of $A$, the construction aligns the left and right boundaries with the $\Phi$-split and $\Phi^{\Downarrow}$-split decompositions after normalization. It provides an explicit boundary-value description, including a closed-form value function $ (r,s,t) \mapsto \dfrac{\theta_{d-r-1}-\theta_t}{\theta_{d-r}-\theta_{t+1}}$ for the polynomial-type case and specializes to the $q$-Racah family with $\theta_i = a + bq^i + cq^{-i}$, yielding a concrete $q$-dependent expression. The results bridge combinatorial Billiard Arrays with the algebraic structure of Leonard systems, clarifying how eigenstructure controls boundary decompositions and enabling explicit normalizations in the $q$-Racah setting.
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 $\{θ_i\}_{i=0}^d$ 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 $θ_i$-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 $Φ=(A;\{E_i\}_{i=0}^d;A^*;\{E_i^*\}_{i=0}^d)$ where $E_i$ is the primitive idempotent of $A$ corresponding to $θ_i$ 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 $Φ$-split (resp. $Φ^{\Downarrow}$-split) decomposition of $V$.