Title:

On Majorana algebras and representations

The basic concepts of Majorana theory were introduced by A. A. Ivanov (2009) as a tool to examine the subalgebras of the Griess algebra V_M from an elementary axiomatic perspective. A Majorana algebra is a commutative nonassociative real algebra generated by a finite set of idempotents, called Majorana axes, that satisfy some properties of Conway's 2Aaxes of V_M. If G is a finite group generated by a Gstable set of involutions T, a Majorana representation of (G,T) is an algebra representation of G on a Majorana algebra V together with a compatible bijection between T and a set of Majorana axes of V . Ivanov's definitions were inspired by Sakuma's theorem, which establishes that any twogenerated Majorana algebra is isomorphic to one of the NortonSakuma algebras. Since then, the construction of Majorana representations of various finite groups has given nontrivial information about the structure of V_M. This thesis concerns two main themes within Majorana theory. The first one is related with the study of some lowdimensional Majorana algebras: the NortonSakuma algebras and the Majorana representations of the symmetric group of degree 4 of shapes (2A,3C) and (2B,3C). For each one of these algebras, all the idempotents, automorphism groups, and maximal associative subalgebras are described. The second theme is related with a Majorana representation V of the alternating group of degree 12 generated by 11,880 Majorana axes. In particular, the possible linear relations between the 3A, 4A, and 5Aaxes of V and the Majorana axes of V are explored. Using the known subalgebras and the inner product structure of V , it is proved that neither sets of 3Aaxes nor 4Aaxes is contained in the linear span of the Majorana axes. When V is a subalgebra of V_M, these results, enhanced with information about the characters of the Monster group, establish that the dimension of V lies between 3,960 and 4,689.
