874 research outputs found
Positive Geometries and Differential Forms with Non-Logarithmic Singularities I
Positive geometries encode the physics of scattering amplitudes in flat
space-time and the wavefunction of the universe in cosmology for a large class
of models. Their unique canonical forms, providing such quantum mechanical
observables, are characterised by having only logarithmic singularities along
all the boundaries of the positive geometry. However, physical observables have
logarithmic singularities just for a subset of theories. Thus, it becomes
crucial to understand whether a similar paradigm can underlie their structure
in more general cases. In this paper we start a systematic investigation of a
geometric-combinatorial characterisation of differential forms with
non-logarithmic singularities, focusing on projective polytopes and related
meromorphic forms with multiple poles. We introduce the notions of covariant
forms and covariant pairings. Covariant forms have poles only along the
boundaries of the given polytope; moreover, their leading Laurent coefficients
along any of the boundaries are still covariant forms on the specific boundary.
Whereas meromorphic forms in covariant pairing with a polytope are associated
to a specific (signed) triangulation, in which poles on spurious boundaries do
not cancel completely, but their order is lowered. These meromorphic forms can
be fully characterised if the polytope they are associated to is viewed as the
restriction of a higher dimensional one onto a hyperplane. The canonical form
of the latter can be mapped into a covariant form or a form in covariant
pairing via a covariant restriction. We show how the geometry of the higher
dimensional polytope determines the structure of these differential forms.
Finally, we discuss how these notions are related to Jeffrey-Kirwan residues
and cosmological polytopes.Comment: 47 pages, figures in Tik
Algorithms for Highly Symmetric Linear and Integer Programs
This paper deals with exploiting symmetry for solving linear and integer
programming problems. Basic properties of linear representations of finite
groups can be used to reduce symmetric linear programming to solving linear
programs of lower dimension. Combining this approach with knowledge of the
geometry of feasible integer solutions yields an algorithm for solving highly
symmetric integer linear programs which only takes time which is linear in the
number of constraints and quadratic in the dimension.Comment: 21 pages, 1 figure; some references and further comments added, title
slightly change
Coxeter submodular functions and deformations of Coxeter permutahedra
We describe the cone of deformations of a Coxeter permutahedron, or
equivalently, the nef cone of the toric variety associated to a Coxeter
complex. This family of polytopes contains polyhedral models for the
Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and
associahedra. Our description extends the known correspondence between
generalized permutahedra, polymatroids, and submodular functions to any finite
reflection group.Comment: Minor edits. To appear in Advances of Mathematic
Brick polytopes, lattice quotients, and Hopf algebras
This paper is motivated by the interplay between the Tamari lattice, J.-L.
Loday's realization of the associahedron, and J.-L. Loday and M. Ronco's Hopf
algebra on binary trees. We show that these constructions extend in the world
of acyclic -triangulations, which were already considered as the vertices of
V. Pilaud and F. Santos' brick polytopes. We describe combinatorially a natural
surjection from the permutations to the acyclic -triangulations. We show
that the fibers of this surjection are the classes of the congruence
on defined as the transitive closure of the rewriting rule for letters
and words on . We then
show that the increasing flip order on -triangulations is the lattice
quotient of the weak order by this congruence. Moreover, we use this surjection
to define a Hopf subalgebra of C. Malvenuto and C. Reutenauer's Hopf algebra on
permutations, indexed by acyclic -triangulations, and to describe the
product and coproduct in this algebra and its dual in term of combinatorial
operations on acyclic -triangulations. Finally, we extend our results in
three directions, describing a Cambrian, a tuple, and a Schr\"oder version of
these constructions.Comment: 59 pages, 32 figure
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