| 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. Recently, there appeared in the literature three results characterizing Q-polynomial distance-regular graphs. In this dissertation, we obtain abstract versions of these characterizations that apply to Leonard pairs. Our first characterization involves the notion of a tail. Our second characterization involves a set of parameters ai. Our third characterization is about a class of Leonard pairs said to be bipartite. In all three cases, the original theorems appeared to rely on the combinatorics of a distance-regular graph. Each of our characterization theorems of Leonard pairs is purely algebraic in nature. |
