105 research outputs found
Counting factorizations of Coxeter elements into products of reflections
In this paper, we count factorizations of Coxeter elements in well-generated
complex reflection groups into products of reflections. We obtain a simple
product formula for the exponential generating function of such factorizations,
which is expressed uniformly in terms of natural parameters of the group. In
the case of factorizations of minimal length, we recover a formula due to P.
Deligne, J. Tits and D. Zagier in the real case and to D. Bessis in the complex
case. For the symmetric group, our formula specializes to a formula of D. M.
Jackson.Comment: 38 pages, including 18 pages appendix. To appear in Journal of the
London Mathematical Society. v3: minor changes and corrected references; v2:
added extended discussion on the definition of Coxeter element
Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
When W is a finite reflection group, the noncrossing partition lattice NCP_W
of type W is a rich combinatorial object, extending the notion of noncrossing
partitions of an n-gon. A formula (for which the only known proofs are
case-by-case) expresses the number of multichains of a given length in NCP_W as
a generalised Fuss-Catalan number, depending on the invariant degrees of W. We
describe how to understand some specifications of this formula in a case-free
way, using an interpretation of the chains of NCP_W as fibers of a
Lyashko-Looijenga covering (LL), constructed from the geometry of the
discriminant hypersurface of W. We study algebraically the map LL, describing
the factorisations of its discriminant and its Jacobian. As byproducts, we
generalise a formula stated by K. Saito for real reflection groups, and we
deduce new enumeration formulas for certain factorisations of a Coxeter element
of W.Comment: 18 pages. Version 2 : corrected typos and improved presentation.
Version 3 : corrected typos, added illustrated example. To appear in Journal
of Algebraic Combinatoric
"Case-free" derivation for Weyl groups of the number of reflection factorisations of a Coxeter element
Chapuy and Stump have given a nice generating series for the number of
factorisations of a Coxeter element as a product of reflections. Their method
is to evaluate case by case a character-theoretic expression. The goal of this
note is to give a uniform evaluation of their character-theoretic expression in
the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig
representations.Comment: 5 page
Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements
We define parabolic quasi-Coxeter elements in well generated complex
reflection groups. We characterize them in multiple natural ways, and we study
two combinatorial objects associated with them: the collections
of reduced reflection factorizations of and
of the relative generating sets of . We compute
the cardinalities of these sets for large families of parabolic quasi-Coxeter
elements and, in particular, we relate the size
with geometric invariants of Frobenius manifolds. This paper is second in a
series of three; we will rely on many of its results in part III to prove
uniform formulas that enumerate full reflection factorizations of parabolic
quasi-Coxeter elements, generalizing the genus- Hurwitz numbers.Comment: v2: 50 pages, minor edits, comments very much welcome
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