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For a finite Coxeter group and a Coxeter element of ; the -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of . Its maximal cones are naturally
indexed by the -sortable elements of . The main result of this paper is that the known bijection cl between -sortable elements and -clusters induces a combinatorial isomorphism of fans. In particular, the -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized
associahedron for . The rays...
A bipolynomial is a holomorphic mapping of a sphere onto a sphere such that some point on
the target sphere has exactly two preimages. The topological invariants of spaces of
bipolynomials without multiple roots are connected with characteristic classes of
rational functions with two poles and generalized braid groups associated to extended
affine Weyl groups of the serie . We prove that the cohomology rings of the spaces of
bipolynomials of bidegree stabilize as tends to infinity and that...
We consider both standard and twisted actions of a (real) Coxeter group on the complement to the complexified reflection hyperplanes by combining the reflections with complex
conjugation. We introduce a natural geometric class of special involutions in and give explicit formulae which describe both actions on the total cohomology in terms of these involutions. As a corollary we prove that the corresponding twisted representation is regular only for
the symmetric group , the Weyl groups...
We compute the Coxeter polynomial of a family of Salem trees, and also the limit of the spectral radii of their Coxeter transformations as the number of their vertices tends to infinity. We also prove that if z is a root of multiplicities for the Coxeter polynomials of the trees respectively, then z is a root for the Coxeter polynomial of their join, of multiplicity at least where .
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