Automorphism groups of lattices, Minkowski reduction domain, and well-rounded retract

04/29/2014 - 12:15pm
04/29/2014 - 1:10pm
Lenny Fukshansky (Claremont McKenna College)

Consider the set of lattices in a Euclidean space which are closed under the natural action of some subgroup of the symmetric group. What are the geometric properties of such lattices? More specifically, how often should one expect lattices like these to have many short vectors? Questions like this come up naturally in lattice theory and can be interpreted geometrically in terms of dimension of the intersection of a certain deformation retract in the space of lattices with facets of the Minkowski domain, a convex polyhedral cone parameterizing arithmetic equivalence classes of quadratic forms. We address these questions for the case of lattices closed under the action of cyclic subgroups of symmetric group, discussing a somewhat surprising phenomenon. This is joint work with X. Sun.

Mudd Science Library 126, Pomona College