Permutation invariant 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 [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

Lattices invariant under the affine general linear group

AbstractIntegral lattices invariant under the affine group AGLm(pt) in its natural permutation module Λ of dimension n=mt are studied. A complete description of such lattices is given. As a consequence we have results on automorphism groups of affine invariant codes over fields and finite residue rings Z/pkZ

Positive representations of finite groups in Riesz spaces

In this paper, which is part of a study of positive representations of locally compact groups in Banach lattices, we initiate the theory of positive representations of finite groups in Riesz spaces. If such a representation has only the zero subspace and possibly the space itself as invariant principal bands, then the space is Archimedean and finite dimensional. Various notions of irreducibility of a positive representation are introduced and, for a finite group acting positively in a space with sufficiently many projections, these are shown to be equal. We describe the finite dimensional positive Archimedean representations of a finite group and establish that, up to order equivalence, these are order direct sums, with unique multiplicities, of the order indecomposable positive representations naturally associated with transitive $G$-spaces. Character theory is shown to break down for positive representations. Induction and systems of imprimitivity are introduced in an ordered context, where the multiplicity formulation of Frobenius reciprocity turns out not to hold.Comment: 23 pages. To appear in International Journal of Mathematic

Permutations of Massive Vacua

We discuss the permutation group G of massive vacua of four-dimensional gauge theories with N=1 supersymmetry that arises upon tracing loops in the space of couplings. We concentrate on superconformal N=4 and N=2 theories with N=1 supersymmetry preserving mass deformations. The permutation group G of massive vacua is the Galois group of characteristic polynomials for the vacuum expectation values of chiral observables. We provide various techniques to effectively compute characteristic polynomials in given theories, and we deduce the existence of varying symmetry breaking patterns of the duality group depending on the gauge algebra and matter content of the theory. Our examples give rise to interesting field extensions of spaces of modular forms.Comment: 44 pages, 1 figur