Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder; Alexander P. Veselov

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arXiv · math/0311190

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder·Alexander P. Veselov

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Abstract

We consider both standard and twisted action of a (real) Coxeter group G on the complement M_G to the complexified reflection hyperplanes by combining the reflections with complex conjugation. We introduce a natural geometric class of special involutions in G and give explicit formulae which describe both actions on the total cohomology H(M_G,C) in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for the symmetric group S_n, the Weyl groups of type D_{2m+1}, E_6 and dihedral groups I_2 (2k+1) and that the standard action has no anti-invariants. We discuss also the relations with the cohomology of generalised braid groups.

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BibTeX RIS

Giovanni Felder, Alexander P. Veselov. 2004-02-06. Coxeter group actions on the complement of hyperplanes and special involutions. https://arxiv.org/abs/math/0311190

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