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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. ItĂ´, 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.

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
Full Text: DOI
[1] Ito, T.; Tanabe, K.; Terwilliger, P., Some algebra related to P- and Q-polynomial association schemes, (), 167-192 · Zbl 0995.05148
[2] Terwilliger, P., Two relations that generalize the q-Serre relations and the dolan-grady relations, (), 377-398 · Zbl 1061.16033
[3] 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
[4] P. Terwilliger, R. Vidunas, Leonard pairs and the Askey-Wilson relations, J. Algebra Appl. in press · Zbl 1062.33018
[5] Zhedanov, A.S., Hidden symmetry of Askey-Wilson polynomials, Teoret. mater. fiz., 89, 190-204, (1991) · Zbl 0744.33009
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