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Tridiagonal pairs and the Askey–Wilson relations. (English) Zbl 1068.05072
Summary: The notion of a tridiagonal pair was introduced by T. Itô, K. Tanabe and P. Terwilliger [DIAMACS, Ser. Discrete Math. Theor. Comput. Sci. 56, 167–192 (2001; Zbl 0995.05148)]. Let $$V$$ denote a nonzero finite-dimensional vector space over a field $${\mathcal F}$$. A tridiagonal pair on $$V$$ is a pair $$(A, A^*)$$, where $$A:V\to V$$ and $$A^*: V\to V$$ are linear transformations that satisfy some conditions. Assume $$(A, A^*)$$ is a tridiagonal pair on $$V$$. Recently P. Terwilliger and R. Vidunas [J. Algebra Appl. 3, 411–426 (2004; Zbl 1062.33018)] showed that if $$A$$ is multiplicity-free on $$V$$, then $$(A, A^*)$$ satisfy the following “Askey-Wilson relation” for some scalars $$\beta$$, $$\gamma$$, $$\gamma^*$$, $$\varrho$$, $$\varrho^*$$, $$\omega$$, $$\eta$$, $$\eta^*$$. $\begin{gathered} A^2 A^*-\beta AA^*A+ A^*A^2- \gamma(AA^*+ A^*A)- \varrho A^*= \gamma^* A^2+ \omega A+\eta I,\\ A^{*2}A- \beta A^* AA^*+ AA^{*2}- \gamma^*(A^* A+ AA^*)- \varrho^* A=\gamma A^{*2}+\omega A^*+ \eta^* I.\end{gathered}$ In this paper, we show that, if a tridiagonal pair $$(A, A^*)$$ satisfy the Askey-Wilson relations, then the eigenspaces of $$A$$ and the eigenspaces of $$A^*$$ have one common dimension, and moreover if $${\mathcal F}$$ is algebraically closed then that common dimension is 1.

##### MSC:
 05E35 Orthogonal polynomials (combinatorics) (MSC2000) 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 05E30 Association schemes, strongly regular graphs
##### Keywords:
Leonard pair; eigenspaces
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##### References:
  Ito, T.; Tanabe, K.; Terwilliger, P., Some algebra related to P- and Q-polynomial association schemes, (), 167-192 · Zbl 0995.05148  Terwilliger, P., Two relations that generalize the q-Serre relations and the dolan-grady relations, (), 377-398 · Zbl 1061.16033  Terwilliger, P., Two linear transformations each tridiagonal with respect to an eigenbasis of the other, Linear algebra appl., 330, 149-203, (2001) · Zbl 0980.05054  P. Terwilliger, R. Vidunas, Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. in press · Zbl 1062.33018  Zhedanov, A.S., Hidden symmetry of Askey-Wilson polynomials, Teoret. mater. fiz., 89, 190-204, (1991) · Zbl 0744.33009
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