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Introduction to Leonard pairs. (English) Zbl 1035.05103
Author’s abstract: In this survey paper we give an elementary introduction to the theory of Leonard pairs. A Leonard pair is defined as follows. Let $$\mathbb{K}$$ denote a field and let $$V$$ denote a vector space over $$\mathbb{K}$$ with finite positive dimension. By a Leonard pair on $$V$$ we mean an ordered pair of linear transformations $$A: V\to V$$ and $$B: V\to V$$ that satisfy conditions (i), (ii) below. (i) There exists a basis for $$V$$ with respect to which the matrix representing $$A$$ is irreducible tridiagonal and the matrix representing $$B$$ is diagonal. (ii) There exists a basis for $$V$$ with respect to which the matrix representing $$A$$ is diagonal and the matrix representing $$B$$ is irreducible tridiagonal.
We give several examples of Leonard pairs and illustrate the use of Leonard pairs in representation theory, combinatorics, and the theory of orthogonal polynomials.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05E35 Orthogonal polynomials (combinatorics) (MSC2000) 17B37 Quantum groups (quantized enveloping algebras) and related deformations 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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