$2A$-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 $A_{12}$, for this might eventually lead to a new and independent construction of the Monster group. In this paper we prove that $A_{12}$ has a unique Majorana representation on the set of its involutions of type $2^2$ and $2^6$ (that is the involutions that fall into the class of Fischer involutions when $A_{12}$ is embedded in the Monster) and we determine the degree and the decomposition into irreducibles of such representation. As a consequence we get that Majorana algebras affording a $2A$-representation of $A_{12}$ and of the Harada-Norton sporadic simple group satisfy the Straight Flush Conjecture. As a by-product we also determine the degree and the decomposition into irreducibles of the Majorana representation induced on the $A_8$ subgroup of $A_{12}$. We finally state a conjecture about Majorana representations of the alternating groups $A_n$, $8\leq n\leq 12$.