34,887 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
Direct topological factorization for topological flows
This paper considers the general question of when a topological action of a
countable group can be factored into a direct product of a nontrivial actions.
In the early 1980's D. Lind considered such questions for -shifts
of finite type. We study in particular direct factorizations of subshifts of
finite type over and other groups, and -subshifts
which are not of finite type. The main results concern direct factors of the
multidimensional full -shift, the multidimensional -colored chessboard
and the Dyck shift over a prime alphabet.
A direct factorization of an expansive -action must be finite,
but a example is provided of a non-expansive -action for which
there is no finite direct prime factorization. The question about existence of
direct prime factorization of expansive actions remains open, even for
.Comment: 21 pages, some changes and remarks added in response to suggestions
by the referee. To appear in ETD
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