Leonard triples and hypercubes

2007/05/03 by Štefko Miklavič, Stefko Miklavic, Miklavic, Stefko · 2 citations
Engineering · Mathematics · #05E30 #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Rings and Algebras (math.RA) #graph theory and CDMA systems #math.CO #math.RA #msc:05E30

paper · pdf · doi:10.48550/arxiv.0705.0518

26 pages

openalex publication_date 2007/05/03 · arxiv created 2008/04/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let V denote a vector space over C with finite positive dimension. By a \em 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 denote the adjacency matrix of Q_D. Fix x ∈ X and let A^* denote the corresponding dual adjacency matrix. Let T denote the subalgebra of Mat_X(C) generated by A, A^*. We refer to T as the \em Terwilliger algebra of Q_D \em with respect to x. The matrices A and A^* are related by the fact that 2 \im A = A^* A^e - A^e A^* and 2 \im A^* = A^e A - A A^e, where 2 \im A^e = A A^* - A^* A and \im²=-1. We show that the triple A, A^*, A^e acts on each irreducible T-module as a Leonard triple. We give a detailed description of these Leonard triples.

Citations

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• “Hidden symmetry” of Askey-Wilson polynomials
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other
• Two linear transformations each tri-diagonal with respect to an eigenbasis of the other; the TD-D canonical form and the LB-UB canonical form
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other; comments on the parameter array

Cited by

• Hypercubes, Leonard triples and the anticommutator spin algebra
• Verma modules and finite-dimensional irreducible modules of the universal Askey–Wilson algebra at roots of unity

Related

• Hypercubes, Leonard triples and the anticommutator spin algebra
• Leonard pairs from 24 points of view
• The classification of Leonard triples of QRacah type
• Two linear transformations each tridiagonal with respect to an eigenbasis of the other
• Leonard pairs having LB-TD form