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1. CJM 2011 (vol 63 pp. 1038)
Critical Points and Resonance of Hyperplane Arrangements If $\Phi_\lambda$ is a master function corresponding to a hyperplane arrangement
$\mathcal A$ and a collection of weights $\lambda$, we investigate the relationship
between the critical set of $\Phi_\lambda$, the variety defined by the vanishing
of the one-form $\omega_\lambda=\operatorname{d} \log \Phi_\lambda$, and the resonance of $\lambda$.
For arrangements satisfying certain conditions, we show that if $\lambda$ is
resonant in dimension $p$, then the critical set
of $\Phi_\lambda$ has codimension
at most $p$. These include all free arrangements and all rank $3$ arrangements.
Keywords:hyperplane arrangement, master function, resonant weights, critical set Categories:32S22, 55N25, 52C35 |
2. CJM 2009 (vol 61 pp. 904)
The Face Semigroup Algebra of a Hyperplane Arrangement This article presents a study of an algebra spanned by the faces of a
hyperplane arrangement. The quiver with relations of the algebra is
computed and the algebra is shown to be a Koszul algebra.
It is shown that the algebra depends only on the intersection lattice of
the hyperplane arrangement. A complete system of primitive orthogonal
idempotents for the algebra is constructed and other algebraic structure
is determined including: a description of the projective indecomposable
modules, the Cartan invariants, projective resolutions of the simple
modules, the Hochschild homology and cohomology, and the Koszul dual
algebra. A new cohomology construction on posets is introduced, and it is
shown that the face semigroup algebra is isomorphic to the cohomology
algebra when this construction is applied to the intersection lattice of
the hyperplane arrangement.
Categories:52C35, 05E25, 16S37 |
3. CJM 2001 (vol 53 pp. 1121)
Monotone Paths on Zonotopes and Oriented Matroids Monotone paths on zonotopes and the natural generalization to maximal
chains in the poset of topes of an oriented matroid or arrangement of
pseudo-hyperplanes are studied with respect to a kind of local move,
called polygon move or flip. It is proved that any monotone path on a
$d$-dimensional zonotope with $n$ generators admits at least $\lceil
2n/(n-d+2) \rceil-1$ flips for all $n \ge d+2 \ge 4$ and that for any
fixed value of $n-d$, this lower bound is sharp for infinitely many
values of $n$. In particular, monotone paths on zonotopes which admit
only three flips are constructed in each dimension $d \ge 3$.
Furthermore, the previously known 2-connectivity of the graph of
monotone paths on a polytope is extended to the 2-connectivity of the
graph of maximal chains of topes of an oriented matroid. An
application in the context of Coxeter groups of a result known to be
valid for monotone paths on simple zonotopes is included.
Categories:52C35, 52B12, 52C40, 20F55 |