A characterization of Leonard pairs using the parameters {a(i)}(i=0)(d)

Author:

Hanson, Edward  

Journal:

LINEAR ALGEBRA AND ITS APPLICATIONS

Issue Date:

2013

Abstract(summary):

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A : V -> V and A* : V -> V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Arlene Pascasio recently obtained a characterization of the Q-polynomial distance-regular graphs using the intersection numbers a(i). In this paper, we extend her results to a linear algebraic level and obtain a characterization of Leonard pairs. Pascasio's argument appears to rely on the underlying combinatorial assumptions, so we take a different approach that is algebraic in nature. (C) 2012 Published by Elsevier Inc.

Page:

2289---2305