Leonard pairs of Racah and Krawtchouk type in LB-TD form
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Let $\Fa$ denote an algebraically closed field with characteristic not two. Fix an integer $d\geq 3$, let $\mathrm{Mat}_{d+1}(\Fa)$ denote the $\Fa$-algebra of $(d+1)\times(d+1)$ matrices with entries in $\Fa$. An ordered pair of matrices $A$, $A^*$ in $\mathrm{Mat}_{d+1}(\Fa)$ is said to be LB-TD form whenever $A$ is lower bidiagonal with subdiagonal entries all $1$ and $A^*$ is irreducible tridiagonal. Let $A$, $A^*$ be a Leonard pair in $\mathrm{Mat}_{d+1}(\Fa)$ with fundamental parameter $\beta =2$, with this assumption there are four families of Leonard pairs, Racah, Hahn, dual Hahn, Krawtchouk type. In this paper we show from these four families only Racah and Krawtchouk have LB-TD form.
Abstract
Let $\Fa$ denote an algebraically closed field with characteristic not two. Fix an integer $d\geq 3$, let $\mathrm{Mat}_{d+1}(\Fa)$ denote the $\Fa$-algebra of $(d+1)\times(d+1)$ matrices with entries in $\Fa$. An ordered pair of matrices $A$, $A^*$ in $\mathrm{Mat}_{d+1}(\Fa)$ is said to be LB-TD form whenever $A$ is lower bidiagonal with subdiagonal entries all $1$ and $A^*$ is irreducible tridiagonal. Let $A$, $A^*$ be a Leonard pair in $\mathrm{Mat}_{d+1}(\Fa)$ with fundamental parameter $\beta =2$, with this assumption there are four families of Leonard pairs, Racah, Hahn, dual Hahn, Krawtchouk type. In this paper we show from these four families only Racah and Krawtchouk have LB-TD form.
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