  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
$A_n$, $8\leq n\leq 12$.
