Basic spin representations of alternating groups, Gow lattices, and Barnes–Wall lattices

In a recent paper, R. Gow showed that in certain cases the basic spin representations of the group $2\mathfrak{A}_n$ (of degree $2^{[\frac{n}{2}]-1}$) can be rational. In such cases, the $2\mathfrak{A}_n$-invariant lattices $\Lambda$ in the corresponding rational module have many interesting properties. In the present paper all possibilities are found for the groups $G=\operatorname{Aut}(\Lambda)$. Also, a conjecture of Gow is proved: For $n=8k$, $ k\in\mathbb{N}$, there is among the $2\mathfrak{A}_n$-invariant lattices the even unimodular Barnes–Wall lattice $BW_{2^{4k-1}}$. At the same time, the rationality of the basic spin representation of $ 2\mathfrak{A}_{8k}$ and the reducibility of $\Lambda/2\Lambda$ as a $2\mathfrak{A}_{8k}$-module are proved.