6 research outputs found
Coxeter factorizations with generalized Jucys-Murphy weights and Matrix Tree theorems for reflection groups
We prove universal (case-free) formulas for the weighted enumeration of
factorizations of Coxeter elements into products of reflections valid in any
well-generated reflection group , in terms of the spectrum of an associated
operator, the -Laplacian. This covers in particular all finite Coxeter
groups. The results of this paper include generalizations of the Matrix Tree
and Matrix Forest theorems to reflection groups, and cover reduced (shortest
length) as well as arbitrary length factorizations.
Our formulas are relative to a choice of weighting system that consists of
free scalar parameters and is defined in terms of a tower of parabolic
subgroups. To study such systems we introduce (a class of) variants of the
Jucys-Murphy elements for every group, from which we define a new notion of
`tower equivalence' of virtual characters. A main technical point is to prove
the tower equivalence between virtual characters naturally appearing in the
problem, and exterior products of the reflection representation of .
Finally we study how this -Laplacian matrix we introduce can be used in
other problems in Coxeter combinatorics. We explain how it defines analogues of
trees for and how it relates them to Coxeter factorizations, we give new
numerological identities between the Coxeter number of and those of its
parabolic subgroups, and finally, when is a Weyl group, we produce a new,
explicit formula for the volume of the corresponding root zonotope.Comment: 57 pages, comments are very much welcom
In which it is proven that, for each parabolic quasi-Coxeter element in a finite real reflection group, the orbits of the Hurwitz action on its reflection factorizations are distinguished by the two obvious invariants
We prove that two reflection factorizations of a parabolic quasi-Coxeter
element in a finite Coxeter group belong to the same Hurwitz orbit if and only
if they generate the same subgroup and have the same multiset of conjugacy
classes. As a lemma, we classify the finite Coxeter groups for which every
reflection generating set that is minimal under inclusion is also of minimum
size.Comment: 10 pages, 1 figure, comments very much welcome
The Generalized Cluster Complex: Refined Enumeration of Faces and Related Parking Spaces
The generalized cluster complex was introduced by Fomin and Reading, as a
natural extension of the Fomin-Zelevinsky cluster complex coming from finite
type cluster algebras. In this work, to each face of this complex we associate
a parabolic conjugacy class of the underlying finite Coxeter group. We show
that the refined enumeration of faces (respectively, positive faces) according
to this data gives an explicit formula in terms of the corresponding
characteristic polynomial (equivalently, in terms of Orlik-Solomon exponents).
This characteristic polynomial originally comes from the theory of hyperplane
arrangements, but it is conveniently defined via the parabolic Burnside ring.
This makes a connection with the theory of parking spaces: our results
eventually rely on some enumeration of chains of noncrossing partitions that
were obtained in this context. The precise relations between the formulas
counting faces and the one counting chains of noncrossing partitions are
combinatorial reciprocities, generalizing the one between Narayana and Kirkman
numbers
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