Generalized quadrangles and subconstituent algebra ¹

• Fernando Levstein ^[1] ; Carolina Maldonado ^[2]
  1. [1] Universidad Nacional de Córdoba

     Universidad Nacional de Córdoba

     Argentina

     
  2. [2] Universidade Federal de Pernambuco

     Universidade Federal de Pernambuco

     Brasil

     
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 12, Nº. 2, 2010, págs. 53-75
• Idioma: inglés
• DOI: 10.4067/S0719-06462010000200005
• Enlaces
  • Texto completo
• Resumen
  • español

    El grafo de puntos de un cuadrángulo generalizado GQ(s, t) es un grafo fuertemente regular G= srg(?, ?, ?, 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 G ).

  • English

    The point graph of a generalized quadrangle GQ (s, t) is a strongly regular graph G = srg( ?, ?, ?, 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 G ).

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