Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
Quiver combinatorics and triangulations of cyclic polytopes
View through CrossRef
Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver. We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2d-dimensional triangulations without interior (d+1)-simplices. This implies that these triangulations form a connected subgraph of the flip graph. Our second result shows how the quiver of a triangulation can be used to identify mutable internal d-simplices. This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.
Title: Quiver combinatorics and triangulations of cyclic polytopes
Description:
Motivated by higher homological algebra, we associate quivers to triangulations of even-dimensional cyclic polytopes and prove two results showing what information about the triangulation is encoded in the quiver.
We first show that the cut quivers of Iyama and Oppermann correspond precisely to 2d-dimensional triangulations without interior (d+1)-simplices.
This implies that these triangulations form a connected subgraph of the flip graph.
Our second result shows how the quiver of a triangulation can be used to identify mutable internal d-simplices.
This points towards what a theory of higher-dimensional quiver mutation might look like and gives a new way of understanding flips of triangulations of even-dimensional cyclic polytopes.
Related Results
Higher-dimensional cluster combinatorics and representation theory
Higher-dimensional cluster combinatorics and representation theory
Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of cla...
Neighborly and almost neighborly configurations, and their duals
Neighborly and almost neighborly configurations, and their duals
This thesis presents new applications of Gale duality to the study of polytopes with extremal combinatorial properties. It consists in two parts. The first one is devoted to the co...
Triangulations of cyclic polytopes
Triangulations of cyclic polytopes
We give a new description of the combinatorics of triangulations of even-dimensional cyclic polytopes, and of their bistellar flips. We show that the tropical exchange relation gov...
Abstract Chiral Polytopes
Abstract Chiral Polytopes
Abstract polytopes are partially ordered sets that satisfy some key aspects of the face lattices of convex polytopes. They are chiral if they have maximal symmetry by combinatorial...
Factorization structures, cones, and polytopes
Factorization structures, cones, and polytopes
Abstract
Factorization structures occur in toric differential and discrete geometry and can be viewed in multiple ways, e.g., as objects determining substantial classes of expli...
Protein kinase activities in rat pancreatic islets of Langerhans
Protein kinase activities in rat pancreatic islets of Langerhans
1. Protein kinase activities in homogenates of rat islets of Langerhans were studied. 2. On incubation of homogenates with [gamma-32P]ATP, incorporation of 32P into protein occurre...
On the bond polytope
On the bond polytope
Abstract
While the maximum cut problem and its corresponding polytope has received a lot of attention inliterature, comparably little is known about the natural clos...
The geometry of Niggli reduction:BGAOL–embedding Niggli reduction and analysis of boundaries
The geometry of Niggli reduction:BGAOL–embedding Niggli reduction and analysis of boundaries
Niggli reduction can be viewed as a series of operations in a six-dimensional space derived from the metric tensor. An implicit embedding of the space of Niggli-reduced cells in a ...