Classical Leonard pairs having LB-TD form

Abstract

ABSTRACTLet denote an algebraically closed field, and let V denote a vector space over with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations on V such that, for each of , there exists a basis for V with respect to which the matrix representing that linear transformation is diagonal and the matrix representing the other linear transformation is irreducible tridiagonal. Assume that the dimension of V is at least 4. In this paper, we consider a class of Leonard pairs said to be of classical type that consists of four families: Racah type, Hahn type, dual-Hahn type and Krawtchouk type. Let be a classical Leonard pair on V . By using Askey-Wilson relations of Leonard pairs and representation theory of universal enveloping algebra , we show that is of either Racah type or Krawtchouk type if and only if there exists a basis for V with respect to which the matrices representing are respectively lower bidiagonal with subdiagonal entries all 1 and irreducible tridiagonal.