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Philipps-Universität Marburg
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Abstract
For $n\geq 3$, let
$\Omega_n$ be the set of line segments between the vertices of a convex $n$-gon.
For $j\geq 2$, a $j$-crossing is a set of $j$ line segments pairwise
intersecting in the relative interior of the $n$-gon. For $k\geq 1$, let
$\Delta_{n,k}$ be the simplicial complex of (type-A) generalized triangulations, i.e.
the simplicial complex of subsets of $\Omega_n$ not containing
any $(k+1)$-crossing.
The complex $\Delta_{n,k}$ has been the central object of numerous papers.
Here we continue this work by considering the complex of type-B generalized
triangulations. For this we identify line-segments in $\Omega_{2n}$ which
can be transformed into each other by a $180^\circ$-rotation of the
$2n$-gon. Let $\F_n$ be the set $\Omega_{2n}$ after identification, then the
complex $\D_{n,k}$ of type-B generalized triangulations
is the simplicial complex of subsets of $\F_n$ not containing
any $(k+1)$-crossing in the above sense. For $k = 1$, we have that $\D_{n,1}$ is
the simplicial complex of type-B triangulations of the $2n$-gon as defined in
\cite{Si} and decomposes into a join of an
$(n-1)$-simplex and the boundary of the $n$-dimensional cyclohedron. We
demonstrate that $\D_{n,k}$ is a pure, $k(n-k)-1+kn$ dimensional complex
that decomposes into a $kn-1$-simplex and a $k(n-k)-1$ dimensional homology sphere.
For $k=n-2$ we show that this homology-sphere is in fact the boundary of a
cyclic polytope. We provide a lower and an upper bound for the number of
maximal faces of $\D_{n,k}$.
On the algebraical side we give a term-order on the monomials in the variables $X_{ij}, 1\leq i,j\leq n$,
such that the corresponding initial ideal of the determinantal ideal
generated by the $(k+1)$ times $(k+1)$ minors of the generic $n \times n$
matrix contains the Stanley-Reisner ideal of $\D_{n,k}$. We show that the minors
form a Gr\"obner-Basis whenever $k\in\{1,n-2,n-1\}$ thereby proving the
equality of both ideals and the
unimodality of the $h$-vector of the determinantal ideal in these cases.
We conjecture this result to be true for all values of $k
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Dates
Created: 2006Issued: 2006-08-16Updated: 2011-08-10
Faculty
Fachbereich Mathematik und Informatik
Publisher
Philipps-Universität Marburg
Language
ger
Data types
DoctoralThesis
Keywords
Symmetrische verallgemeinerte TriangulierungDeterminantielles IdealDeterminantal idealTyp-B TriangulierungGeneralized triangulation
DFG-subjects
Monomiales IdealKonvexes PolytopGr?r-BasisTriangulierung
DDC-Numbers
510
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Soll, Daniel: Polytopale Konstruktionen in der Algebra. : Philipps-Universität Marburg 2006-08-16. DOI: https://doi.org/10.17192/z2006.0137.
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This item has been published with the following license: In Copyright