Leonard pairs having specified end-entries

Abstract

Fix an algebraically closed field $\mathbb{F}$ and an integer $d\ge 3$. Let $V$ be a vector space over $\mathbb{F}$ with dimension $d+1$. A Leonard pair on $V$ is an ordered pair of diagonalizable linear transformations $A:V\to V$ and $A^{\ast }:V\to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let $\{v_i{\}}_{i=0}^d$ (resp.\ $\{v_i^{\ast }{\}}_{i=0}^d$) be such an eigenbasis for $A$ (resp.\ $A^{\ast }$). For $0\le i\le d$ define a linear transformation $E_i:V\to V$ such that $E_i{v}_i=v_i$ and $E_i{v}_j=0$ if $j\ne i$ $(0\le j\le d)$. Define $E_i^{\ast }:V\to V$ in a similar way. The sequence $\Phi =(A,\{E_i{\}}_{i=0}^d,A^{\ast },\{E_i^{\ast }{\}}_{i=0}^d)$ is called a Leonard system on $V$ with diameter $d$. With respect to the basis $\{v_i{\}}_{i=0}^d$, let $\{{þ}_i{\}}_{i=0}^d$ (resp.\ $\{a_i^{\ast }{\}}_{i=0}^d$) be the diagonal entries of the matrix representing $A$ (resp.\ $A^{\ast }$). With respect to the basis $\{v_i^{\ast }{\}}_{i=0}^d$, let $\{{\theta }_i^{\ast }{\}}_{i=0}^d$ (resp.\ $\{a_i{\}}_{i=0}^d$) be the diagonal entries of the matrix representing $A^{\ast }$ (resp.\ $A$). It is known that $\{{\theta }_i{\}}_{i=0}^d$ (resp. $\{{þ}_i^{\ast }{\}}_{i=0}^d$) are mutually distinct, and the expressions $({\theta }_{i-1}-{\theta }_{i+2})/({\theta }_i-{\theta }_{i+1})$, $({\theta }_{i-1}^{\ast }-{\theta }_{i+2}^{\ast })/({\theta }_i^{\ast }-{\theta }_{i+1}^{\ast })$ are equal and independent of $i$ for $1\le i\le d-2$. Write this common value as $\beta +1$. In the present paper we consider the "end-entries" ${\theta }_0$, ${\theta }_d$, ${\theta }_0^{\ast }$, ${\theta }_d^{\ast }$, $a_0$, $a_d$, $a_0^{\ast }$, $a_d^{\ast }$. We prove that a Leonard system with diameter $d$ is determined up to isomorphism by its end-entries and $\beta$ if and only if either (i) $\beta \ne \pm 2$ and $q^{d-1}\ne -1$, where $\beta =q+q^{-1}$, or (ii) $\beta =\pm 2$ and $\text{Char}(\mathbb{F})\ne 2$.

arXiv admin note: substantial text overlap with arXiv:1408.2180

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