The 2A-Majorana representations of the Harada-Norton group

We show that all $2A$-Majorana representations of the Harada-Norton group $F_5$ have the same shape. If ${\mathcal R}$ is such a representation, we determine, using the theory of association schemes, the dimension and the irreducible constituents of the linear span $U$ of the Majorana axes. Finally, we prove that, if ${\mathcal R}$ is based on the (unique) embedding of $F_5$ in the Monster, $U$ is closed under the algebra product.