Permutation invariant lattices

Authors

Lenny Fukshansky, Claremont McKenna CollegeFollow
Stephan Ramon Garcia, Pomona CollegeFollow
Xun Sun, Claremont Graduate UniversityFollow

Student Co-author

CGU Graduate

Document Type

Article - preprint

Department

Mathematics (Pomona), Mathematics (CMC), Mathematical Sciences (CGU)

Publication Date

2015

Keywords

automorphism groups of lattices, well-rounded lattices, cyclic lattices

Abstract

We say that a Euclidean lattice in Rⁿ 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 T E S_n, 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 [1]. 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.

Comments

Final published version can be found at: Fukshansky, L., Garcia, S.R., Sun, X., Permutation invariant lattices, Discrete Mathematics 338 (2015), 1536-1541.

Rights Information

© 2015 Elsevier B.V.

DOI

10.1016/j.disc.2015.03.016

Recommended Citation

Fukshansky, L., Garcia, S.R., Sun, X., Permutation invariant lattices, Discrete Mathematics 338 (2015), 1536-1541.