2,058 research outputs found
On Lattice-Free Orbit Polytopes
Given a permutation group acting on coordinates of , we
consider lattice-free polytopes that are the convex hull of an orbit of one
integral vector. The vertices of such polytopes are called \emph{core points}
and they play a key role in a recent approach to exploit symmetry in integer
convex optimization problems. Here, naturally the question arises, for which
groups the number of core points is finite up to translations by vectors fixed
by the group. In this paper we consider transitive permutation groups and prove
this type of finiteness for the -homogeneous ones. We provide tools for
practical computations of core points and obtain a complete list of
representatives for all -homogeneous groups up to degree twelve. For
transitive groups that are not -homogeneous we conjecture that there exist
infinitely many core points up to translations by the all-ones-vector. We prove
our conjecture for two large classes of groups: For imprimitive groups and
groups that have an irrational invariant subspace.Comment: 27 pages, 2 figures; with minor adaptions according to referee
comments; to appear in Discrete and Computational Geometr
Momentum polytopes of projective spherical varieties and related K\"ahler geometry
We apply the combinatorial theory of spherical varieties to characterize the
momentum polytopes of polarized projective spherical varieties. This enables us
to derive a classification of these varieties, without specifying the open
orbit, as well as a classification of all Fano spherical varieties. In the
setting of multiplicity free compact and connected Hamiltonian manifolds, we
obtain a necessary and sufficient condition involving momentum polytopes for
such manifolds to be K\"ahler and classify the invariant compatible complex
structures of a given K\"ahler multiplicity free compact and connected
Hamiltonian manifold.Comment: v1: 32 pages. v2: 47 pages, fixed errors, improved exposition,
expanded Section 7. v3: 47 pages, implemented changes and corrections
requested by refere
Toric topology
We survey some results on toric topology.Comment: English translation of the Japanese article which appeared in
"Sugaku" vol. 62 (2010), 386-41
Specht Polytopes and Specht Matroids
The generators of the classical Specht module satisfy intricate relations. We
introduce the Specht matroid, which keeps track of these relations, and the
Specht polytope, which also keeps track of convexity relations. We establish
basic facts about the Specht polytope, for example, that the symmetric group
acts transitively on its vertices and irreducibly on its ambient real vector
space. A similar construction builds a matroid and polytope for a tensor
product of Specht modules, giving "Kronecker matroids" and "Kronecker
polytopes" instead of the usual Kronecker coefficients. We dub this process of
upgrading numbers to matroids and polytopes "matroidification," giving two more
examples. In the course of describing these objects, we also give an elementary
account of the construction of Specht modules different from the standard one.
Finally, we provide code to compute with Specht matroids and their Chow rings.Comment: 32 pages, 5 figure
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