Permutation invariant lattices

We say that a Euclidean lattice in  Rn is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group  Sn, i.e., if the lattice is closed under the action of some non-identity elements of  Sn. Given a fixed element τ∈Sn, we study properties of the set of all lattices closed under the action of  τ: we call such lattices τ-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio (2002, 2007), which we previously studied in Fukshansky and Sun (2014). Continuing our investigation, we discuss some basic properties of permutation invariant lattices, in particular proving that the subset of well-rounded lattices in the set of all τ-invariant lattices in  Rn has positive co-dimension (and hence comprises zero proportion) for all  τ different from an n-cycle.