The Terwilliger algebra of a distance-regular graph of negative type

Miklavič, Štefko

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Title:	The Terwilliger algebra of a distance-regular graph of negative type
Authors:	ID Miklavič, Štefko (Author)
Files:	http://dx.doi.org/10.1016/j.laa.2008.07.013
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	Let ▫$\Gamma$▫ denote a distance-regular graph with diameter ▫$D \ge 3$▫. Assume ▫$\Gamma$▫ has classical parameters ▫$(D,b,\alpha,\beta)▫$ with ▫$b < -1$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫ and let ▫$A \in {\mathrm{Mat}}_X(\mathbb{C})$▫ denote the adjacency matrix of ▫$\Gamma$▫. 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 call ▫$T$▫ the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. We show that up to isomorphism there exist exactly two irreducible ▫$T$▫-modules with endpoint 1; their dimensions are ▫$D$▫ and ▫$2D-2$▫. For these ▫$T$▫-modules we display a basis consisting of eigenvectors for ▫$A^\ast$▫, and for each basis we give the action of ▫$A$▫.
Keywords:	distance-regular graph, negative type, Terwilliger algebra
Year of publishing:	2009
Number of pages:	str. 251-270
Numbering:	Vol. 430, no. 1
PID:	20.500.12556/RUP-172
ISSN:	0024-3795
UDC:	519.1
COBISS.SI-ID:	2132965
Publication date in RUP:	15.10.2013
Views:	3736
Downloads:	110
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