Article - preprint
Mathematics (Pomona), Mathematics (CMC), Mathematical Sciences (CGU)
automorphism groups of lattices, well-rounded lattices, cyclic 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 T E Sn, we study properties of the set of all lattices closed under the action of T: we call such lattices T-invariant. These lattices naturally generalize cyclic lattices introduced by Micciancio in [7,8], 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 latices in the set of all T-invariant lattices in Rn has positive co-dimension (and hence comprises zero proportion) for all T different from an n-cycle.
© 2015 Elsevier B.V.
Fukshansky, L., Garcia, S.R., Sun, X., Permutation invariant lattices, Discrete Mathematics 338 (2015), 1536-1541.