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

Abstract : 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.
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https://hal.archives-ouvertes.fr/hal-00365536
Contributor : Lucas Vienne Connect in order to contact the contributor
Submitted on : Tuesday, March 3, 2009 - 4:37:58 PM
Last modification on : Wednesday, October 20, 2021 - 3:18:46 AM
Long-term archiving on: : Tuesday, June 8, 2010 - 11:06:11 PM

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  • HAL Id : hal-00365536, version 1
  • ARXIV : 0903.0912

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

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