620 research outputs found
The uniform face ideals of a simplicial complex
We define the uniform face ideal of a simplicial complex with respect to an
ordered proper vertex colouring of the complex. This ideal is a monomial ideal
which is generally not squarefree. We show that such a monomial ideal has a
linear resolution, as do all of its powers, if and only if the colouring
satisfies a certain nesting property.
In the case when the colouring is nested, we give a minimal cellular
resolution supported on a cubical complex. From this, we give the graded Betti
numbers in terms of the face-vector of the underlying simplicial complex.
Moreover, we explicitly describe the Boij-S\"oderberg decompositions of both
the ideal and its quotient. We also give explicit formul\ae\ for the
codimension, Krull dimension, multiplicity, projective dimension, depth, and
regularity. Further still, we describe the associated primes, and we show that
they are persistent.Comment: 34 pages, 8 figure
Linear syzygies, hyperbolic Coxeter groups and regularity
We show that the virtual cohomological dimension of a Coxeter group is
essentially the regularity of the Stanley--Reisner ring of its nerve. Using
this connection between geometric group theory and commutative algebra, as well
as techniques from the theory of hyperbolic Coxeter groups, we study the
behavior of the Castelnuovo--Mumford regularity of square-free quadratic
monomial ideals. We construct examples of such ideals which exhibit arbitrarily
high regularity after linear syzygies for arbitrarily many steps. We give a
doubly logarithmic bound on the regularity as a function of the number of
variables if these ideals are Cohen--Macaulay.Comment: 22 pages, v2: final version as in Compositio Mat
The toric h-vector of a cubical complex in terms of noncrossing partition statistics
This paper introduces a new and simple statistic on noncrossing partitions
that expresses each coordinate of the toric -vector of a cubical complex,
written in the basis of the Adin -vector entries, as the total weight of all
noncrossing partitions. The same model may also be used to obtain a very simple
combinatorial interpretation of the contribution of a cubical shelling
component to the toric -vector. In this model, a strengthening of the
symmetry expressed by the Dehn-Sommerville equations may be derived from the
self-duality of the noncrossing partition lattice, exhibited by the involution
of Simion and Ullman
Manifolds of isospectral arrow matrices
An arrow matrix is a matrix with zeroes outside the main diagonal, first row,
and first column. We consider the space of Hermitian arrow
-matrices with fixed simple spectrum . We prove
that this space is a smooth -manifold, and its smooth structure is
independent on the spectrum. Next, this manifold carries the locally standard
torus action: we describe the topology and combinatorics of its orbit space. If
, the orbit space is not a polytope, hence
this manifold is not quasitoric. However, there is a natural permutation action
on which induces the combined action of a semidirect product
. The orbit space of this large action is a simple
polytope. The structure of this polytope is described in the paper.
In case , the space is a solid torus with
boundary subdivided into hexagons in a regular way. This description allows to
compute the cohomology ring and equivariant cohomology ring of the
6-dimensional manifold using the general theory developed by
the first author. This theory is also applied to a certain -dimensional
manifold called the twin of . The twin carries a
half-dimensional torus action and has nontrivial tangent and normal bundles.Comment: 29 pages, 8 figure
The combinatorics of hyperbolized manifolds
A topological version of a longstanding conjecture of H. Hopf, originally
proposed by W. Thurston, states that the sign of the Euler characteristic of a
closed aspherical manifold of dimension depends only on the parity of
. Gromov defined several hyperbolization functors which produce an
aspherical manifold from a given simplicial or cubical manifold. We investigate
the combinatorics of several of these hyperbolizations and verify the Euler
Characteristic Sign Conjecture for each of them. In addition, we explore
further combinatorial properties of these hyperbolizations as they relate to
several well-studied generating functions
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