2009/03/05 by Lucas Vienne, Vienne, Lucas
Mathematics · #05C25 #05C50 #05C62 #20B05 #20B20 #20B25 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #math.CO #math.GR #msc:05C25 #msc:05C50 #msc:05C62 #msc:20B05 #msc:20B20 #msc:20B25
paper · pdf · doi:10.48550/arxiv.0903.0912
arxiv created 2009/03/05 · arxiv updated 2009/12/01
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=Ui | 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 ui of the lines Ui (i in X) such that forall i,j in X, ||ui||=1, and if i is different from j then (ui|uj)=\gvei,j.c, with \gvei,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 \gvei,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=PSL2(q) on a set of q+1 equiangular lines in a vector space whose dimension is (q+1)/2.