Permutation invariant lattices

Abstract

We say that a Euclidean lattice in ${\mathbb{R}}^n$ is permutation invariant if its automorphism group has non-trivial intersection with the symmetric group $S_n$, i.e., if the lattice is closed under the action of some non-identity elements of $S_n$. Given a fixed element $\tau \in S_n$, we study properties of the set of all lattices closed under the action of $\tau$: we call such lattices $\tau$-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio, which we studied in a recent paper. 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 $\tau$-invariant lattices in ${\mathbb{R}}^n$ has positive co-dimension (and hence comprises zero proportion) for all $\tau$ different from an $n$-cycle.

corrected Lemma 2.1

References in corpus (1)

• Well-rounded equivariant deformation retracts of Teichmüller spaces

Cited by in corpus (2)

• On lattices generated by finite Abelian groups
• On the geometry of cyclic lattices