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Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

For a finite Coxeter group W and a Coxeter element c of W ; the c -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of W . Its maximal cones are naturally indexed by the c -sortable elements of W . The main result of this paper is that the known bijection cl c between c -sortable elements and c -clusters induces a combinatorial isomorphism of fans. In particular, the c -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for W . The rays...

Cohomology rings of spaces of generic bipolynomials and extended affine Weyl groups of serie A

Fabien Napolitano (2003)

Annales de l’institut Fourier

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 A . We prove that the cohomology rings of the spaces of bipolynomials of bidegree ( k , l ) stabilize as k tends to infinity and that...

Coxeter group actions on the complement of hyperplanes and special involutions

Giovanni Felder, A. Veselov (2005)

Journal of the European Mathematical Society

We consider both standard and twisted actions of a (real) Coxeter group G on the complement 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 * ( 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...

Coxeter polynomials of Salem trees

Charalampos A. Evripidou (2015)

Colloquium Mathematicae

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 m , . . . , m k for the Coxeter polynomials of the trees , . . . , k respectively, then z is a root for the Coxeter polynomial of their join, of multiplicity at least m i n m - m , . . . , m - m k where m = m + + m k .

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