Saturated Majorana representations of A_12

Majorana representations have been introduced by Ivanov in order to provide an axiomatic framework for studying the actions on the Griess algebra of the Monster and of its subgroups generated by Fischer involutions. A crucial step in this programme is to obtain an explicit description of the Majorana representations of A12, the largest alternating group admitting a Majorana representation, for this might eventually lead to a new and independent construction of the Monster group. In this paper we prove that A12 has two possible Majorana sets, one of which is the set of bitraspositions, the other is the union of the set of bitranspositions with the set of fix-point-free involutions. The latter case (the saturated case) is most interesting, since the Majorana set is precisely the set of involutions of A_12 that fall into the class of Fischer involutions when A_12 is embedded in the Monster. We prove that A_12 has unique saturated Majorana representation and we determine its degree and decomposition into irreducibles. As consequences we get that the Harada-Norton group has, up to equivalence, a unique Majorana representation and every Majorana algebra, affording either a Majorana representation of the Harada-Norton group or a saturated Majorana representation of A_12, satisfies the Straight Flush Conjecture. As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on A_8, the four point stabilizer subgroup of A_12. We finally state a conjecture about Majorana representations of the alternating groups A_n, 8 ≤ n ≤ 12.