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Published in:
2012 | OriginalPaper | Chapter
Arrangements stable under the Coxeter groups
Authors : Hidehiko Kamiya, Akimichi Takemura, Hiroaki Terao
Published in: Configuration Spaces
Publisher: Scuola Normale Superiore
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Let
B
be a real hyperplane arrangement which is stable under the action of a Coxeter group
W
. Then
W
acts naturally on the set of chambers of
B
. We assume that
B
is disjoint from the Coxeter arrangement
A
=
A
(W)
of
W.
In this paper, we show that the W-orbits of the set of chambers of
B
are in one-to-one correspondence with the chambers of C =
B
∪
B
which are contained in an arbitrarily fixed chamber of
A
. From this fact, we find that the number of W-orbits of the set of chambers of
B
is given by the number of chambers of C divided by the order of
W.
We will also study the set of chambers of C which are contained in a chamber
b
of
B
. We prove that the cardinality of this set is equal to the order of the isotropy subgroup
W
b
of
b.
We illustrate these results with some examples, and solve an open problem in [H. Kamiya, A. Takemura, H. Terao, Ranking patterns of unfolding models of codimension one, Adv. in Appl. Math. 47 (2011) 379–400] by using our results.