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Counting chambers in restricted Coxeter arrangements

  • Tilman Möller
  • Gerhard RöhrleEmail author
Article
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Abstract

Solomon showed that the Poincaré polynomial of a Coxeter group W satisfies a product decomposition depending on the exponents of W. This polynomial coincides with the rank-generating function of the poset of regions of the underlying Coxeter arrangement. In this note we determine all instances when the analogous factorization property of the rank-generating function of the poset of regions holds for a restriction of a Coxeter arrangement. It turns out that this is always the case with the exception of some instances in type \(E_8\).

Keywords

Coxeter arrangement Restriction of a Coxeter arrangement Poset of regions of a real arrangement Factorization of rank-generating function 

Mathematics Subject Classification

20F55 52B30 52C35 14N20 

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Notes

Acknowledgements

We are grateful to T. Hoge for checking that the simplicial arrangement “\(A_4(17)\)” from Grünbaum’s list coincides with the restriction \((E_8,A_2 A_3)\). We would also like to thank C. Stump for helpful discussions concerning computations in SAGE. The research of this work was supported by DFG-Grant RO 1072/16-1.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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