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Affine transformations of a Leonard pair. (English) Zbl 1143.05337

Summary: Let \(\mathbb K\) denote a field and let \(V\) denote a vector space over \(\mathbb K\) with finite positive dimension. We consider an ordered pair of linear transformations \(A : V \to V\) and \(A^* : V \to V\) that satisfy (i) and (ii) below: 6mm
(i)
There exists a basis for \(V\) with respect to which the matrix representing \(A\) is irreducible tridiagonal and the matrix representing \(A^*\) is diagonal.
(ii)
There exists a basis for \(V\) with respect to which the matrix representing \(A^*\) is irreducible tridiagonal and the matrix representing \(A\) is diagonal.
We call such a pair a Leonard pair on \(V\). Let \(\xi\), \(\zeta\), \(\xi^*\), \(\zeta^*\) denote scalars in \(\mathbb K\) with \(\xi\), \(\xi^*\) nonzero, and note that \(\xi A + \zeta I\), \(\xi^*A^* + \zeta^*I\) is a Leonard pair on \(V\). We give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair \(A,A^*\). We also give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair \(A^*,A\).

MSC:

05E35 Orthogonal polynomials (combinatorics) (MSC2000)
05E30 Association schemes, strongly regular graphs
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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