A characterization of Leonard pairs using the parameters $\{a_i{\}}_{i=0}^d$

Abstract

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A:V\to V$ and $A^{\ast }:V\to V$ 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 $A^{\ast }$ is diagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^{\ast }$ 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.

21 pages. arXiv admin note: substantial text overlap with arXiv:0911.0098

References in corpus (2)

• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array
• Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form

Cited by in corpus (1)

• How to recognize a Leonard pair