Leonard triples and hypercubes

Miklavič, Štefko

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Title:	Leonard triples and hypercubes
Authors:	ID Miklavič, Štefko (Author)
Files:	http://dx.doi.org/10.1007/s10801-007-0108-x
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
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:	mathematics, graph theory, Leonard triple, distance-regular graph, hypercube, Terwilliger algebra
Year of publishing:	2007
Number of pages:	str. 397-424
Numbering:	Vol. 28, no. 3
PID:	20.500.12556/RUP-1597
ISSN:	0925-9899
UDC:	519.17
COBISS.SI-ID:	14624857
Publication date in RUP:	15.10.2013
Views:	5743
Downloads:	125
Metadata:
:	MIKLAVIČ, Štefko, 2007, Leonard triples and hypercubes. [online]. 2007. Vol. 28, no. 3, p. 397–424. [Accessed 28 March 2025]. Retrieved from: http://dx.doi.org/10.1007/s10801-007-0108-x Copy citation