2013/08/18 by Edward Hanson, Hanson, Edward
Mathematics · #05E30 (Secondary) #15A21 (Primary) #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:05E30 #msc:15A21
paper · pdf · doi:10.48550/arxiv.1308.3826
21 pages. arXiv admin note: substantial text overlap with arXiv:1205.4368
arxiv created 2013/08/18 · arxiv updated 2013/08/20
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.