A characterization of 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 . In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.

Totally bipartite tridiagonal pairs

“…Each of the elements in ( 52) is fixed by †, and therefore contained in (M A * M ) sym . The elements (52)…”

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

“…In [2,3], we extended existing characterizations of Q-polynomial distance-regular graphs to obtain characterizations of Leonard pairs. We mention some details.…”

“…In [4, Theorem 1.1], these tails were used to characterize Q-polynomial distance-regular graphs. In [2,Definition 4.5], we introduced an abstract version of the tail notion and in [2, Theorem 5.1], we used it to characterize Leonard pairs. In [7,Theorem 1.2], the a i were used to characterize Q-polynomial distanceregular graphs.…”

“…In [5, Theorem 1.1], Lang used tails to characterize bipartite Q-polynomial distance-regular graphs. We use our abstract version of the tail notion [2,Definition 4.5] to characterize bipartite Leonard pairs. Whereas Lang's arguments relied on the combinatorics of a distance-regular graph, our results are purely algebraic in nature.…”