Leonard triples and hypercubes

Abstract

Let ▫$V$▫ denote a vector space over ▫$\mathbb{C}$▫ 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 ▫${\mathcal{Q}}_D$▫ denote the graph of the ▫$D$▫-dimensional hypercube. Let ▫$X$ denote the vertex set of ▫${\mathcal{Q}}_D$▫ and let ▫$A \in {\mathrm{Mat}}_X ({\mathbb{C}})$▫ denote the adjacency matrix of ▫${\mathcal{Q}}_D$▫. Fix ▫$x \in X$▫ and let ▫$A^\ast \in {\mathrm{Mat}}_X({\mathbb{C}})$▫ denote the corresponding dual adjacency matrix. Let ▫$T$▫ denote the subalgebra of ▫${\mathrm{Mat}}_X({\mathbb{C}})$ generated by ▫$A,A^\ast$▫. We refer to ▫$T$▫ as the Terwilliger algebra of ▫${\mathcal{Q}}_D$▫ with respect to ▫$x$▫. The matrices ▫$A$▫ and ▫$A^\ast$▫ are related by the fact that ▫$2iA = A^\ast A^\varepsilon - A^\varepsilon A^\ast$▫ and ▫$2iA^\ast = A^\varepsilon A - AA^\varepsilon$▫, where ▫$2iA^\varepsilon = AA^\ast - A^\ast A$▫ and ▫$i^2 = -1$▫. We show that the triple ▫$A$▫, ▫$A^\ast$▫, ▫$A^\varepsilon$▫ acts on each irreducible ▫$T$▫-module as a Leonard triple. We give a detailed description of these Leonard triples.

Keywords

matematika;teorija grafov;razdaljno regularni grafi;Leonardova trojica;hiperkocka;Terwilligerjeva algebra;mathematics;graph theory;Leonard triple;distance-regular graph;hypercube;Terwilliger algebra;

Data

Language:	English
Year of publishing:	2007
Typology:	1.01 - Original Scientific Article
Organization:	UP - University of Primorska
UDC:	519.17
COBISS:	14624857
ISSN:	0925-9899
Views:	3698
Downloads:	121
Average score:	0 (0 votes)
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Other data

Secondary language:	English
Secondary keywords:	matematika;teorija grafov;razdaljno regularni grafi;Leonardova trojica;hiperkocka;Terwilligerjeva algebra;
Type (COBISS):	Not categorized
Pages:	str. 397-424
Volume:	ǂVol. ǂ28
Issue:	ǂno. ǂ3
Chronology:	2008
DOI:	10.1007/s10801-007-0108-x
ID:	1473509