Groupes d'isométries permutant doublement transitivement un ensemble de droites vectorielles

Lucas Vienne (1)

1 LAREMA - Laboratoire Angevin de Recherche en Mathématiques

Let X be a non-empty finite set, E be a finite dimensional euclidean vector space and G a finite subgroup of O(E), the orthognal group of E. Suppose GG={U_i | i in X } is a finite set of linear lines in E and an orbit of G on which its operation is twice transitive. Then GG is an equiangular set of lines, which means that we can find a real number ''c'', and generators u_i of the lines U_i (i in X) such that forall i,j in X, ||u_i||=1 , and if i is different from j then (u_i|u_j)=\gve_{i,j}.c, with \gve_{i,j} in {-1,+1\} Let Gamma be the simple graph whose set of vertices is X, two of them, say i and j, being linked when \gve_{i,j} = -1. In this article we first explore the relationship between double transitivity of G and geometric properties of Gamma. Then we construct several graphs associated with a twice transitive group G, in particular any of Paley's graphs is associated with a representation of G=PSL_2(q) on a set of q+1 equiangular lines in a vector space whose dimension is (q+1)/2.

Mots clés: groupe; graphe; fortement regulier; strongly regular graphs; groupe doublement transitif; Paley graphs; equiangular lines

Domaines: Combinatoire [math.CO]; Mathématiques générales [math.GM]; Théorie des groupes [math.GR]

Lucas Vienne. Groupes d'isométries permutant doublement transitivement un ensemble de droites vectorielles. 2009. ⟨hal-00365536⟩