Groupes d'isom\'etries permutant doublement transitivement un ensemble de droites vectorielles

TL;DR

This paper investigates the geometric and combinatorial properties of finite groups acting twice transitively on sets of equiangular lines, establishing connections with graph theory and group representations.

Contribution

It explores the relationship between double transitivity of groups and geometric properties of associated graphs, and constructs specific graphs linked to group representations.

Findings

Equiangular lines form when a group acts twice transitively.

Graphs associated with these groups include Paley's graphs.

Representation of PSL_2(q) relates to equiangular lines in a vector space.

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.