Leonard triples and hypercubes

• Authors: Štefko Miklavič
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2008

Let V denote a vector space over 驴 with finite positive dimension. By a Leonard triple on V we mean an ordered triple of linear operators on V such that for each of these operators there exists a basis of V with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let D denote a positive integer and let Q D denote the graph of the D-dimensional hypercube. Let X denote the vertex set of Q D and let $A\in {\rm Mat}_{X}({\mathbb{C}})$ denote the adjacency matrix of Q D . Fix x驴X and let $A^{*}\in {\rm Mat}_{X}({\mathbb{C}})$ denote the corresponding dual adjacency matrix. Let T denote the subalgebra of ${\rm Mat}_{X}({\mathbb{C}})$ generated by A,A *. We refer to T as the Terwilliger algebra of Q D with respect to x. The matrices A and A * are related by the fact that 2i A=A * A 驴 驴A 驴 A * and 2i A *=A 驴 A驴AA 驴 , where 2i A 驴 =AA *驴A * A and i 2=驴1. We show that the triple A, A *, A 驴 acts on each irreducible T-module as a Leonard triple. We give a detailed description of these Leonard triples.