A characterization of bipartite Leonard pairs using the notion of a tail

Abstract

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 . Very roughly speaking, a Leonard pair is a linear algebraic abstraction of a Q-polynomial distance-regular graph. There is a well-known class of distance-regular graphs said to be bipartite and there is a related notion of a bipartite Leonard pair. Recently, M. S. Lang introduced the notion of a tail for bipartite distance-regular graphs, and there is an abstract version of this tail notion. Lang characterized the bipartite Q-polynomial distance-regular graphs using tails. In this paper, we obtain a similar characterization of the bipartite Leonard pairs using tails. Whereas Lang's arguments relied on the combinatorics of a distance-regular graph, our results are purely algebraic in nature.

Notes on the Leonard System Classification

“…Therefore the inclusion holds with equality. ðiiiÞ ) ðiÞ For 0 i d let V i denote the common value in (7). Observe that…”

“…In the present paper we make use of the fact that the split canonical form still exists under weaker assumptions; these are described in Proposition 7.6 below. • The concept of a normalizing idempotent was introduced by Edward Hanson in [7,Sect. 6].…”

Notes on the Leonard system classification

“…Therefore the inclusion holds with equality. (iii) ⇒ (i) For 0 ≤ i ≤ d let V i denote the common value in (7). Observe that…”