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

TL;DR

This paper characterizes bipartite Leonard pairs, algebraic structures related to distance-regular graphs, using the concept of a tail, providing an algebraic perspective distinct from combinatorial approaches.

Contribution

It offers an algebraic characterization of bipartite Leonard pairs through the notion of a tail, extending previous combinatorial characterizations of related graph structures.

Findings

Bipartite Leonard pairs can be characterized algebraically using tails.

The algebraic approach differs from previous combinatorial methods.

The results connect Leonard pairs with the tail concept in a purely algebraic framework.

Abstract

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^*: V\rightarrow V$ 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.