Introduction to Leonard pairs

In this survey paper we give an elementary introduction to the theory of Leonard pairs. A Leonard pair is defined as follows. Let K denote a field and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A:V→V and B:V→V that satisfy conditions (i), (ii) below. (i) exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal. exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal. ve several examples of Leonard pairs. Using these we illustrate how Leonard pairs arise in representation theory, combinatorics, and the theory of orthogonal polynomials.