Non-rigidity degrees of root lattices and their duals

Abstract

Non-rigidity degree of a lattice $L$, nrd$L$, is dimension of the L-type domain to which $L$ belongs. We complete here the table of nrd's of all root lattices and their duals; namely, the hardest remaining case of $D_n^*$, and the case of $E_7^*$ are decided. We describe explicitly the $L$-type domain ${\cal D}(D_n^*)$, $n \ge 4$. For $n$ odd, it is a non-simplicial polyhedral open cone of dimension $n$. For $n$ even, it is one-dimensional, i.e. for even $n$, $D_n^*$ is an edge form.