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29 March 2024
 
  » arxiv » 1409.4333

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Leonard pairs having specified end-entries
Kazumasa Nomura ;
Date 15 Sep 2014
AbstractFix an algebraically closed field $mathbb{F}$ and an integer $d geq 3$. 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 $A: V o V$ and $A^* : V o V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let ${v_i}_{i=0}^d$ (resp. ${v^*_i}_{i=0}^d$) be such an eigenbasis for $A$ (resp. $A^*$). For $0 leq i leq d$ define a linear transformation $E_i : V o V$ such that $E_i v_i=v_i$ and $E_i v_j =0$ if $j eq i$ $(0 leq j leq d)$. Define $E^*_i : V o V$ in a similar way. The sequence $Phi =(A, {E_i}_{i=0}^d, A^*, {E^*_i}_{i=0}^d)$ is called a Leonard system on $V$ with diameter $d$. With respect to the basis ${v_i}_{i=0}^d$, let ${ h_i}_{i=0}^d$ (resp. ${a^*_i}_{i=0}^d$) be the diagonal entries of the matrix representing $A$ (resp. $A^*$). With respect to the basis ${v^*_i}_{i=0}^d$, let ${ heta^*_i}_{i=0}^d$ (resp. ${a_i}_{i=0}^d$) be the diagonal entries of the matrix representing $A^*$ (resp. $A$). It is known that ${ heta_i}_{i=0}^d$ (resp. ${ h^*_i}_{i=0}^d$) are mutually distinct, and the expressions $( heta_{i-1}- heta_{i+2})/( heta_i- heta_{i+1})$, $( heta^*_{i-1}- heta^*_{i+2})/( heta^*_i - heta^*_{i+1})$ are equal and independent of $i$ for $1 leq i leq d-2$. Write this common value as $eta + 1$. In the present paper we consider the "end-entries" $ heta_0$, $ heta_d$, $ heta^*_0$, $ heta^*_d$, $a_0$, $a_d$, $a^*_0$, $a^*_d$. We prove that a Leonard system with diameter $d$ is determined up to isomorphism by its end-entries and $eta$ if and only if either (i) $eta eq pm 2$ and $q^{d-1} eq -1$, where $eta=q+q^{-1}$, or (ii) $eta = pm 2$ and $ ext{Char}(mathbb{F}) eq 2$.
Source arXiv, 1409.4333
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