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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.2 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000200005 

CUBO A Mathematical Journal Vol.12, N° 02, (53–75). June 2010

 

Generalized quadrangles and subconstituent algebra 1

 

Fernando Levstein and Carolina Maldonado

FaMAF-CIEM,UNC, Universidad Nacional de Córdoba Medina Allende y Haya de la Torre. CP 5000 - Córdoba, Argentina email: levstein@famaf.unc.edu.ar

Departamento de Matemática, Centro de Ciências Exatas e da Natureza, Universidade Federal de Pernambuco Av. Prof. Luiz Freire, s/n Cidade Universitária - Recife, Brasil email: cmaldona@famaf.unc.edu.ar


ABSTRACT

The point graph of a generalized quadrangle GQ (s, t) is a strongly regular graph Γ = srg(𝓥,𝓚,ʎ, μ) whose parameters depend on s and t. By a detailed analysis of the adjacency matrix we compute the Terwilliger algebra of this kind of graphs (and denoted it by 𝓣 ). We find that there are only two non-isomorphic Terwilliger algebras for all the generalized quadrangles. The two classes correspond to wether s2 = t or not. We decompose the algebra into direct sum of simple ideals. Considering the action? × Cx→ Cx we find the decomposition into irreducible 𝓣 -submodules of Cx (where X is the set of vertices of the Γ ).

Key words and phrases: strongly regular graphs, generalized quadrangles, Terwilliger algebra.


RESUMEN

El grafo de puntos de un cuadrángulo generalizado GQ(s, t) es un grafo fuertemente regular Γ= srg(𝓥, 𝓚, ʎ, μ) cuyos parámetros dependen de s y t. Mediante un análisis detallado de la matriz de adyacencia, calculamos el álgebra de Terwilliger (𝓣 -álgebra) de esta familia de grafos. Encontramos que para todos los cuadrángulos generalizados, existen solo dos tipos no isomorfos de 𝓣 -álgebras asociadas. Dichas clases dependen de si s2 = t o no. Descomponemos el álgebra en suma directa de ideales simples. Considerando la acción 𝓣 × Cx→ Cx encontramos la descomposición de Cx en 𝓣 -submódulos irreducibles. (X es el conjunto de vértices de Γ ).

AMS (MOS) Subj. Class.: 05E30


Notas

1This work supported by FACEPE, CCEN UFPE and CIEM-FaMAF UNC, CONICET.

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Received: November 2008.

Revised: February 2009.