On Perfection Relations in Lattices

Let $\Lb$ be a lattice in a Euclidean space $E$, with kissing number $s$ and perfection rank $r$, that is, the rank in $\End^{\text{sym}}(E)$ of the set of orthogonal projections to minimal vectors of $\Lb$. This defines a space of \emph{perfection relations}, of dimension $s-r$. We focus on ``short relations'', in connection with the index theory, previously developed by Watson, Ry\v{s}kov, Zahareva and the second author in [W], [R], [Z] and [M1].