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Leonard pairs from 24 points of view. (English) Zbl 1040.05030
A Leonard pair on a finite-dimensional vector space \(V\) over a field \(\mathbb{K}\) is an ordered pair of linear transformations \(A: V\to V\) and \(A^*: V\to V\) that satisfy the following two conditions. (i) There exists a basis for \(V\) with respect to which the matrix representing \(A\) is diagonal and the matrix representing \(A^*\) is irreducible tridiagonal. (ii) There exists a basis for \(V\) with respect to which the matrix representing \(A^*\) is diagonal and the matrix representing \(A\) is irreducible tridiagonal.
The author motivates the theory of Leonard pairs by their connection with certain orthogonal polynomials, and their use in representation theory of some algebras as well as in the theory of \(P\)-polynomial and \(Q\)-polynomial association schemes, a topic in which the author is the expert. See for instance, from the same author [J. Comput. Appl. Math. 153, No. 1–2, 463–475 (2003; Zbl 1035.05103)].
In this quite extensive paper, the author investigates 24 bases for \(V\) on which the action of \(A\) and \(A^*\) take an “attractive” (sic) form. The relation between these 24 bases is given by the Cayley graph for the symmetric group \(S_4\).

MSC:
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.)
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
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