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? × C^x→ C^x we find the decomposition into irreducible 𝓣 -submodules of C^x (where X is the set of vertices of the Γ ).

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

RESUMEN

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.