74 research outputs found
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
Generalized Goulden-Yong duals and signed minimal factorizations
We show the equivalence between one-way reflections and relative projective
representations. We construct generalized Goulden-Yong duals using reverse
Garside element actions and folded chord diagrams. We give two applications of
the generalized Goulden-Yong duals: constructing generalized Pr\"{u}fer codes
and counting signed factorizations using the matrix-tree theorem.Comment: 36 pages, 15 figure
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
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