59 research outputs found
Faces of Birkhoff Polytopes
The Birkhoff polytope B(n) is the convex hull of all (n x n) permutation
matrices, i.e., matrices where precisely one entry in each row and column is
one, and zeros at all other places. This is a widely studied polytope with
various applications throughout mathematics.
In this paper we study combinatorial types L of faces of a Birkhoff polytope.
The Birkhoff dimension bd(L) of L is the smallest n such that B(n) has a face
with combinatorial type L.
By a result of Billera and Sarangarajan, a combinatorial type L of a
d-dimensional face appears in some B(k) for k less or equal to 2d, so bd(L) is
at most d. We will characterize those types whose Birkhoff dimension is at
least 2d-3, and we prove that any type whose Birkhoff dimension is at least d
is either a product or a wedge over some lower dimensional face. Further, we
computationally classify all d-dimensional combinatorial types for d between 2
and 8.Comment: 29 page
The -Construction for Lattices, Spheres and Polytopes
We describe and analyze a new construction that produces new Eulerian
lattices from old ones. It specializes to a construction that produces new
strongly regular cellular spheres (whose face lattices are Eulerian). The
construction does not always specialize to convex polytopes; however, in a
number of cases where we can realize it, it produces interesting classes of
polytopes. Thus we produce an infinite family of rational 2-simplicial 2-simple
4-polytopes, as requested by Eppstein, Kuperberg and Ziegler. We also construct
for each an infinite family of -simplicial 2-simple
-polytopes, thus solving a problem of Gr\"unbaum.Comment: 21 pages, many figure
Polymake and Lattice Polytopes
The polymake software system deals with convex polytopes and related objects
from geometric combinatorics. This note reports on a new implementation of a
subclass for lattice polytopes. The features displayed are enabled by recent
changes to the polymake core, which will be discussed briefly.Comment: 12 pages, 1 figur
Polytopes associated to Dihedral Groups
In this note we investigate the convex hull of those -permutation
matrices that correspond to symmetries of a regular -gon. We give the
complete facet description. As an application, we show that this yields a
Gorenstein polytope, and we determine the Ehrhart -vector
Bier spheres and posets
In 1992 Thomas Bier presented a strikingly simple method to produce a huge
number of simplicial (n-2)-spheres on 2n vertices as deleted joins of a
simplicial complex on n vertices with its combinatorial Alexander dual.
Here we interpret his construction as giving the poset of all the intervals
in a boolean algebra that "cut across an ideal." Thus we arrive at a
substantial generalization of Bier's construction: the Bier posets Bier(P,I) of
an arbitrary bounded poset P of finite length. In the case of face posets of PL
spheres this yields cellular "generalized Bier spheres." In the case of
Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I)
inherit these properties.
In the boolean case originally considered by Bier, we show that all the
spheres produced by his construction are shellable, which yields "many
shellable spheres", most of which lack convex realization. Finally, we present
simple explicit formulas for the g-vectors of these simplicial spheres and
verify that they satisfy a strong form of the g-conjecture for spheres.Comment: 15 pages. Revised and slightly extended version; last section
rewritte
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