915 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
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}
Primitive filtrations of the modules of invariant logarithmic forms of Coxeter arrangements
We define {\bf primitive derivations} for Coxeter arrangements which may not
be irreducible. Using those derivations, we introduce the {\bf primitive
filtrations} of the module of invariant logarithmic differential forms for an
arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular,
when the Coxeter arrangement is irreducible with a constant multiplicity, the
primitive filtration has already been studied, which generalizes the Hodge
filtration introduced by K. Saito
The Euler multiplicity and addition-deletion theorems for multiarrangements
The addition-deletion theorems for hyperplane arrangements, which were
originally shown in [H. Terao, Arrangements of hyperplanes and their freeness
I, II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 293--320], provide
useful ways to construct examples of free arrangements. In this article, we
prove addition-deletion theorems for multiarrangements. A key to the
generalization is the definition of a new multiplicity, called the Euler
multiplicity, of a restricted multiarrangement. We compute the Euler
multiplicities in many cases. Then we apply the addition-deletion theorems to
various arrangements including supersolvable arrangements and the Coxeter
arrangement of type to construct free and non-free multiarrangements
Equivariant multiplicities of Coxeter arrangements and invariant bases
Let \A be an irreducible Coxeter arrangement and be its Coxeter group.
Then naturally acts on \A. A multiplicity \bfm : \A\rightarrow \Z is
said to be equivariant when \bfm is constant on each -orbit of \A. In
this article, we prove that the multi-derivation module D(\A, \bfm) is a free
module whenever \bfm is equivariant by explicitly constructing a basis, which
generalizes the main theorem of \cite{T02}. The main tool is a primitive
derivation and its covariant derivative. Moreover, we show that the
-invariant part D(\A, \bfm)^{W} for any multiplicity \bfm is a free
module over the -invariant subring
The characteristic polynomial of a multiarrangement
Given a multiarrangement of hyperplanes we define a series by sums of the
Hilbert series of the derivation modules of the multiarrangement. This series
turns out to be a polynomial. Using this polynomial we define the
characteristic polynomial of a multiarrangement which generalizes the
characteristic polynomial of an arragnement. The characteristic polynomial of
an arrangement is a combinatorial invariant, but this generalized
characteristic polynomial is not. However, when the multiarrangement is free,
we are able to prove the factorization theorem for the characteristic
polynomial. The main result is a formula that relates `global' data to `local'
data of a multiarrangement given by the coefficients of the respective
characteristic polynomials. This result gives a new necessary condition for a
multiarrangement to be free. Consequently it provides a simple method to show
that a given multiarrangement is not free.Comment: 12 pages, 2 figure
Increase of cloud cover due to reduced sea ice in the Arctic Ocean in MIROC6 historical simulations
The Tenth Symposium on Polar Science/Ordinary sessions: [OM] Polar Meteorology and Glaciology, Wed. 4 Dec. / Entrance Hall (1st floor) , National Institute of Polar Researc
A primitive derivation and logarithmic differential forms of Coxeter arrangements
Let be a finite irreducible real reflection group, which is a Coxeter
group. We explicitly construct a basis for the module of differential 1-forms
with logarithmic poles along the Coxeter arrangement by using a primitive
derivation. As a consequence, we extend the Hodge filtration, indexed by
nonnegative integers, into a filtration indexed by all integers. This
filtration coincides with the filtration by the order of poles. The results are
translated into the derivation case
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