Search

We say that a Euclidean lattice in 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 τ ∈ S n , 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 (2002, 2007), which we previously studied in Fukshansky and Sun (2014). 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 R n has positive co-dimension (and hence comprises zero proportion) for all τ different from an n -cycle. [ABSTRACT FROM AUTHOR]

Abstract: We produce an explicit parameterization of well-rounded sublattices of the hexagonal lattice in the plane, splitting them into similarity classes. We use this parameterization to study the number, the greatest minimal norm, and the highest signal-to-noise ratio of well-rounded sublattices of the hexagonal lattice of a fixed index. This investigation parallels earlier work by Bernstein, Sloane, and Wright where similar questions were addressed on the space of all sublattices of the hexagonal lattice. Our restriction is motivated by the importance of well-rounded lattices for discrete optimization problems. Finally, we also discuss the existence of a natural combinatorial structure on the set of similarity classes of well-rounded sublattices of the hexagonal lattice, induced by the action of a certain matrix monoid. [ABSTRACT FROM AUTHOR]