14,349 research outputs found
The Projected Faces Property and Polyhedral Relations
Margot (1994) in his doctoral dissertation studied extended formulations of
combinatorial polytopes that arise from "smaller" polytopes via some
composition rule. He introduced the "projected faces property" of a polytope
and showed that this property suffices to iteratively build extended
formulations of composed polytopes.
For the composed polytopes, we show that an extended formulation of the type
studied in this paper is always possible only if the smaller polytopes have the
projected faces property. Therefore, this produces a characterization of the
projected faces property.
Affinely generated polyhedral relations were introduced by Kaibel and
Pashkovich (2011) to construct extended formulations for the convex hull of the
images of a point under the action of some finite group of reflections. In this
paper we prove that the projected faces property and affinely generated
polyhedral relation are equivalent conditions
Reflection groups and polytopes over finite fields, II
When the standard representation of a crystallographic Coxeter group
is reduced modulo an odd prime , a finite representation in some orthogonal
space over is obtained. If has a string diagram, the
latter group will often be the automorphism group of a finite regular polytope.
In Part I we described the basics of this construction and enumerated the
polytopes associated with the groups of rank 3 and the groups of spherical or
Euclidean type. In this paper, we investigate such families of polytopes for
more general choices of , including all groups of rank 4. In
particular, we study in depth the interplay between their geometric properties
and the algebraic structure of the corresponding finite orthogonal group.Comment: 30 pages (Advances in Applied Mathematics, to appear
Realization spaces of 4-polytopes are universal
Let be a -dimensional polytope. The {\em realization space}
of~ is the space of all polytopes that are combinatorially
equivalent to~, modulo affine transformations. We report on work by the
first author, which shows that realization spaces of \mbox{4-dimensional}
polytopes can be ``arbitrarily bad'': namely, for every primary semialgebraic
set~ defined over~, there is a -polytope whose realization
space is ``stably equivalent'' to~. This implies that the realization space
of a -polytope can have the homotopy type of an arbitrary finite simplicial
complex, and that all algebraic numbers are needed to realize all -
polytopes. The proof is constructive. These results sharply contrast the
-dimensional case, where realization spaces are contractible and all
polytopes are realizable with integral coordinates (Steinitz's Theorem). No
similar universality result was previously known in any fixed dimension.Comment: 10 page
Reflection Groups and Polytopes over Finite Fields, III
When the standard representation of a crystallographic Coxeter group is
reduced modulo an odd prime p, one obtains a finite group G^p acting on some
orthogonal space over Z_p . If the Coxeter group has a string diagram, then G^p
will often be the automorphism group of a finite abstract regular polytope. In
parts I and II we established the basics of this construction and enumerated
the polytopes associated to groups of rank at most 4, as well as all groups of
spherical or Euclidean type. Here we extend the range of our earlier criteria
for the polytopality of G^p . Building on this we investigate the class of
3-infinity groups of general rank, and then complete a survey of those locally
toroidal polytopes which can be described by our construction.Comment: Advances in Applied Mathematics (to appear); 19 page
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 classical representation theory have been
increasingly studied. Cyclic polytopes are classical objects of study in convex
geometry. In particular, their triangulations have been studied with a view
towards generalizing the rich combinatorial structure of triangulations of
polygons. In this paper, we demonstrate a connection between these two
seemingly unrelated subjects.
We study triangulations of even-dimensional cyclic polytopes and tilting
modules for higher Auslander algebras of linearly oriented type A which are
summands of the cluster tilting module. We show that such tilting modules
correspond bijectively to triangulations. Moreover mutations of tilting modules
correspond to bistellar flips of triangulations.
For any d-representation finite algebra we introduce a certain d-dimensional
cluster category and study its cluster tilting objects. For higher Auslander
algebras of linearly oriented type A we obtain a similar correspondence between
cluster tilting objects and triangulations of a certain cyclic polytope.
Finally we study certain functions on generalized laminations in cyclic
polytopes, and show that they satisfy analogues of tropical cluster exchange
relations. Moreover we observe that the terms of these exchange relations are
closely related to the terms occuring in the mutation of cluster tilting
objects.Comment: 41 pages. v4: minor corrections throughout the pape
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