Totally almost bipartite Leonard pairs and Leonard triples of q-Racah type

Abstract

Let denote an algebraically closed field of characteristic zero. Let V denote a vector space over with finite positive dimension. A Leonard pair on V is an ordered pair of linear transformations in such that for each of these transformations there exists a basis for V with respect to which the matrix representing that transformation is diagonal and the matrix representing the other transformation is irreducible tridiagonal. Whenever these two tridiagonal matrices are almost bipartite, the Leonard pair is said to be totally almost bipartite. The notion of a Leonard triple and the corresponding notion of totally almost bipartite are similarly defined. Let q denote a quantum parameter of a Leonard pair and let ‘TAB’ be an abbreviation for ‘totally almost bipartite’. In this paper we show that a TAB Leonard pair with q equal to is of Bannai/Ito type, and a TAB Leonard pair with q being not a root of unity is of q-Racah type. Under the assumption that q is not a root of unity, we classify, up to isomorphism, the TAB Leonard pairs of q-Racah type and the TAB Leonard triples of q-Racah type.