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Deformations of Coxeter Hyperplane Arrangements

✍ Scribed by Alexander Postnikov; Richard P. Stanley

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
479 KB
Volume
91
Category
Article
ISSN
0097-3165

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✦ Synopsis

We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x i &x j =1, 1 i< j n, is equal to the number of alternating trees on n+1 vertices. Remarkably, these numbers have several additional combinatorial interpretations in terms of binary trees, partially ordered sets, and tournaments. More generally, we give formulae for the number of regions and the Poincare polynomial of certain finite subarrangements of the affine Coxeter arrangement of type A n&1 . These formulae enable us to prove a ``Riemann hypothesis'' on the location of zeros of the Poincare polynomial. We give asymptotics of the Poincare polynomials when n goes to infinity. We also consider some generic deformations of Coxeter arrangements of type A n&1 .

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