362 research outputs found
Chromatic Ramsey number of acyclic hypergraphs
Suppose that is an acyclic -uniform hypergraph, with . We
define the (-color) chromatic Ramsey number as the smallest
with the following property: if the edges of any -chromatic -uniform
hypergraph are colored with colors in any manner, there is a monochromatic
copy of . We observe that is well defined and where
is the -color Ramsey number of . We give linear upper bounds
for when T is a matching or star, proving that for , and where
and are, respectively, the -uniform matching and star with
edges.
The general bounds are improved for -uniform hypergraphs. We prove that
, extending a special case of Alon-Frankl-Lov\'asz' theorem.
We also prove that , which is sharp for . This is
a corollary of a more general result. We define as the 1-intersection
graph of , whose vertices represent hyperedges and whose edges represent
intersections of hyperedges in exactly one vertex. We prove that for any -uniform hypergraph (assuming ). The proof uses the list coloring version of Brooks' theorem.Comment: 10 page
Positroid cluster structures from relabeled plabic graphs
The Grassmannian is a disjoint union of open positroid varieties ,
certain smooth irreducible subvarieties whose definition is motivated by total
positivity. The coordinate ring of is a cluster algebra, and each reduced
plabic graph for determines a cluster. We study the effect of
relabeling the boundary vertices of by a permutation . Under suitable
hypotheses on the permutation, we show that the relabeled graph
determines a cluster for a different open positroid variety . As a key
step of the proof, we show that and are isomorphic by a nontrivial
twist isomorphism. Our constructions yield many cluster structures on each open
positroid variety , given by plabic graphs with appropriately relabeled
boundary. We conjecture that the seeds in all of these cluster structures are
related by a combination of mutations and Laurent monomial transformations
involving frozen variables, and establish this conjecture for (open) Schubert
and opposite Schubert varieties. As an application, we also show that for
certain reduced plabic graphs , the "source" cluster and the "target"
cluster are related by mutation and Laurent monomial rescalings.Comment: 45 pages, comments welcome! v2: minor change
Saturation of Newton polytopes of type A and D cluster variables
We study Newton polytopes for cluster variables in cluster algebras
of types A and D. A famous property of cluster algebras
is the Laurent phenomenon: each cluster variable can be written as a Laurent
polynomial in the cluster variables of the initial seed . The cluster
variable Newton polytopes are the Newton polytopes of these Laurent
polynomials. We show that if has principal coefficients or boundary
frozen variables, then all cluster variable Newton polytopes are saturated. We
also characterize when these Newton polytopes are \emph{empty}; that is, when
they have no non-vertex lattice points.Comment: 33 Pages, 21 Figures. Second version includes additional results for
the cases of principal coefficients and no frozen variables, as well as
characterizations of emptiness of Newton polytope
The m=2 amplituhedron and the hypersimplex: signs, clusters, triangulations, Eulerian numbers
The hypersimplex is the image of the positive Grassmannian
under the moment map. It is a polytope of dimension
in . Meanwhile, the amplituhedron is the
projection of the positive Grassmannian into
under a map induced by a matrix .
Introduced in the context of scattering amplitudes, it is not a polytope, and
has dimension . Nevertheless, there seem to be remarkable connections
between these two objects via T-duality, as was first noted by
Lukowski--Parisi--Williams (LPW). In this paper we use ideas from oriented
matroid theory, total positivity, and the geometry of the hypersimplex and
positroid polytopes to obtain a deeper understanding of the amplituhedron. We
show that the inequalities cutting out positroid polytopes -- images of
positroid cells of under the moment map -- translate into
sign conditions characterizing the T-dual Grasstopes -- images of positroid
cells of under . Moreover, we subdivide the
amplituhedron into chambers, just as the hypersimplex can be subdivided into
simplices, with both chambers and simplices enumerated by the Eulerian numbers.
We prove the main conjecture of (LPW): a collection of positroid polytopes is a
triangulation of if and only if the collection of T-dual
Grasstopes is a triangulation of for all .
Moreover, we prove Arkani-Hamed--Thomas--Trnka's conjectural sign-flip
characterization of , and
Lukowski--Parisi--Spradlin--Volovich's conjectures on cluster adjacency
and on generalized triangles (images of -dimensional positroid cells which
map injectively into ). Finally, we introduce new
cluster structures in the amplituhedron.Comment: 72 pages, many figures, comments welcome. v4: Minor edits v3:
Strengthened results on triangulations and realizability of amplituhedron
sign chambers. v2: Results added to Section 11.4, minor edit
Braid variety cluster structures, I: 3D plabic graphs
We introduce -dimensional generalizations of Postnikov's plabic graphs and
use them to establish cluster structures for type braid varieties. Our
results include known cluster structures on open positroid varieties and double
Bruhat cells, and establish new cluster structures for type open Richardson
varieties
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