Title:

Majorana representations and the Coxeter Groups G^(m,n,p)

The work presented in this thesis is a contribution to Majorana theory as introduced by A. A. Ivanov in [Iva09]. Inspired by a theorem of S. Sakuma [Sak07], Majorana theory is an axiomatisation of seven properties of the Monster algebra VM, invariant under the Monster group M, and of some of its idempotents. For a finite group G generated by a Ginvariant set T of involutions one can define what it means for G to have a Majorana representation with respect to T. It yields a Ginvariant Majorana algebra X generated by a set A of idempotents called Majorana axes. To each axis a in A an automorphism Tau(a) of X called Majorana involution is associated, so that X, A and Tau(A) satisfy the Majorana axioms. In this thesis we give a review of the Majorana algebras already obtained and we motivate the following objective: the classification of all Majorana algebras V generated by three Majorana axes a_1, a_2 and a_3 such that V also contains a Majorana axis a_1,2 with associated automorphism Tau(a_1,2) := Tau(a_1) Tau(a_2). This objective requires the classification of all subgroups G :=< Tau(a_{i})  i in [1;3]> of GL(V ), which are necessarily quotients of the Coxeter groups G^(m,n,p) := such that m, n, p belong to [1; 6] and the set of conjugates of x, y, xy and z is 6transposition. We prove that all such groups are quotients of only eleven finite groups. Comparing with S. P. Norton's list of `trianglepoint' configurations of 2A involutions in M [Nor85], we observe that nine of them 2Aembed in M. The next step is to answer which of those eleven groups can be generated by Majorana involutions. Using partial results already known, we manage to answer this question completely for one of the eleven groups isomorphic to L2(11). For the remaining ten groups we give technical restrictions on the structure of their possible Majorana algebras, thus laying the foundations for further progress.
