54,360 research outputs found
Ordered Exchange Graphs
The exchange graph of a cluster algebra encodes the combinatorics of
mutations of clusters. Through the recent "categorifications" of cluster
algebras using representation theory one obtains a whole variety of exchange
graphs associated with objects such as a finite-dimensional algebra or a
differential graded algebra concentrated in non-positive degrees. These
constructions often come from variations of the concept of tilting, the
vertices of the exchange graph being torsion pairs, t-structures, silting
objects, support -tilting modules and so on. All these exchange graphs
stemming from representation theory have the additional feature that they are
the Hasse quiver of a partial order which is naturally defined for the objects.
In this sense, the exchange graphs studied in this article can be considered as
a generalization or as a completion of the poset of tilting modules which has
been studied by Happel and Unger. The goal of this article is to axiomatize the
thus obtained structure of an ordered exchange graph, to present the various
constructions of ordered exchange graphs and to relate them among each other.Comment: References updated, and Theorem A.7 adde
Hermitian metrics on F-manifolds
An -manifold is complex manifold with a multiplication on the holomorphic
tangent bundle with a certain integrability condition. Important examples are
Frobenius manifolds and especially base spaces of universal unfoldings of
isolated hypersurface singularities. This paper reviews the construction of
hermitian metrics on -manifolds from geometry. It clarifies the logic
between several notions. It also introduces a new {\it canonical} hermitian
metric. Near irreducible points it makes the manifold almost hyperbolic. This
holds for the singularity case and will hopefully lead to applications there.Comment: 2nd version 36 pages. Compared to the 1st version (32 pages), the
sections 2.4 and 2.5 have been extende
Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth
We give a fixed-parameter tractable algorithm that, given a parameter and
two graphs , either concludes that one of these graphs has treewidth
at least , or determines whether and are isomorphic. The running
time of the algorithm on an -vertex graph is ,
and this is the first fixed-parameter algorithm for Graph Isomorphism
parameterized by treewidth.
Our algorithm in fact solves the more general canonization problem. We namely
design a procedure working in time that, for a
given graph on vertices, either concludes that the treewidth of is
at least , or: * finds in an isomorphic-invariant way a graph
that is isomorphic to ; * finds an isomorphism-invariant
construction term --- an algebraic expression that encodes together with a
tree decomposition of of width .
Hence, the isomorphism test reduces to verifying whether the computed
isomorphic copies or the construction terms for and are equal.Comment: Full version of a paper presented at FOCS 201
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