Nonrigidity Degrees of Root Lattices and their Duals

Abstract

The nonrigidity degree of a lattice L, nrdL, is the dimension of the L-type domain to which L belongs. We complete here the table of nrd's of all irreducible root lattices and their duals (we give also the minimal rank of their Delaunay polytopes). In particular, the hardest remaining case of D n *, and the case of E 7* are decided. As any root lattice is a direct sum of some irreducible ones, its nrd is a sum of nrd's of the summands. We describe explicitly the L-type domain $$\mathcal{D}(D_n^* )$$ , n≥ 4. For n odd, it is a nonsimplicial, polyhedral, open cone of dimension n. For n even, it is one-dimensional, i.e. any D 2m * corresponds to an edge form.