Article - postprint
Mathematics (CMC), Mathematics (Pomona), Mathematical Sciences (CGU)
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 in [8, 9], which we previously studied in . 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.
© 2015 Fukshansky, Garcia, Sun
Fukshansky, L., Garcia, S., Sun, X. Permutation invariant lattices, Discrete Mathematics, vol. 338 no. 8 (2015), pg. 1536--1541.