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Article - postprint


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

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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 [1]. 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.


Source: Author's post-print manuscript in PDF.

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© 2015 Fukshansky, Garcia, Sun

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