Leonard pairs having zero-diagonal TD-TD form

摘要

Fix an algebraically closed field IF and an integer n >= 1. Let Mat(n)(F) denote the F-algebra consisting of the n x n matrices that have all entries in F. We consider a pair of diagonalizable matrices in Mat(n)(F), each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in Mat(n)(F). In the present paper, we find all Leonard pairs A, A* in Mat(n)(F) such that each of A and A* is irreducible tridiagonal with all diagonal entries 0. This solves a problem given by Paul Terwilliger. (C) 2015 Elsevier Inc. All rights reserved.