Saturated Majorana representations of A_12

Majorana representations have been introduced by Ivanov in~\cite{Iva} 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 $A_{12}$ (by \cite{FIM2}, the largest alternating group admitting a Majorana representation) for this might eventually lead to a new and independent construction of the Monster group (see ~\cite[Section 4, pag.115]{IvInd}). In this paper we prove that $A_{12}$ has two possible Majorana sets, one of which is the set $\mathcal X_b$ of involutions of cycle type $2^2$, the other is the union of $\mathcal X_b$ with the set $\mathcal X_s$ of involutions of cycle type $2^6$. 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 (see~\cite{IvCon} and~\cite{IvInd}). 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 $A_n$, $8\leq n\leq 12$.