60 research outputs found
The freeness of Shi-Catalan arrangements
Let be a finite Weyl group and \A be the corresponding Weyl
arrangement. A deformation of \A is an affine arrangement which is obtained
by adding to each hyperplane H\in\A several parallel translations of by
the positive root (and its integer multiples) perpendicular to . We say that
a deformation is -equivariant if the number of parallel hyperplanes of each
hyperplane H\in \A depends only on the -orbit of . We prove that the
conings of the -equivariant deformations are free arrangements under a
Shi-Catalan condition and give a formula for the number of chambers. This
generalizes Yoshinaga's theorem conjectured by Edelman-Reiner.Comment: 12 page
Freeness for restriction arrangements of the extended Shi and Catalan arrangements
The extended Shi and Catalan arrangements are well investigated arrangements.
In this paper, we prove that the cone of the extended Catalan arrangement of
type A is always hereditarily free, while we determine the dimension that the
cone of the extended Shi arrangement of type A is hereditarily free. For this
purpose, using digraphs, we define a class of arrangements which is closed
under restriction and which contains the extended Shi and Catalan arrangements.
We also characterize the freeness for the cone of this arrangement by graphical
conditions.Comment: 18 pages, 8 figure
Freeness of hyperplane arrangements and related topics
This is the expanded notes of the lecture by the author in "Arrangements in
Pyrenees", June 2012. We are discussing relations of freeness and splitting
problems of vector bundles, several techniques proving freeness of hyperplane
arrangements, K. Saito's theory of primitive derivations for Coxeter
arrangements, their application to combinatorial problems and related
conjectures.Comment: 28 page
Simple-root bases for Shi arrangements
In his affirmative answer to the Edelman-Reiner conjecture, Yoshinaga proved
that the logarithmic derivation modules of the cones of the extended Shi
arrangements are free modules. However, all we know about the bases is their
existence and degrees. In this article, we introduce two distinguished bases
for the modules. More specifically, we will define and study the simple-root
basis plus (SRB+) and the simple-root basis minus (SRB-) when a primitive
derivation is fixed. They have remarkable properties relevant to the simple
roots and those properties characterize the bases
Free filtrations of affine Weyl arrangements and the ideal-Shi arrangements
In this article we prove that the ideal-Shi arrangements are free central
arrangements of hyperplanes satisfying the dual-partition formula. Then it
immediately follows that there exists a saturated free filtration of the cone
of any affine Weyl arrangement such that each filter is a free subarrangement
satisfying the dual-partition formula. This generalizes the main result in
\cite{ABCHT} which affirmatively settled a conjecture by Sommers and Tymoczko
\cite{SomTym}
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