26,973 research outputs found
Some Combinatorial Results for Complex Reflection Groups
AbstractIn this paper, we prove that a simple system for a subsystem Ψ of the complex root system Φ can always be chosen as a subset of the positive system Φ+of Φ. Furthermore, we show that a set of distinguished coset representatives can be found for every reflection subgroup of the complex reflection groups. The corresponding results for real crystallographic root systems and their reflection groups (i.e., Weyl groups) are well known (see [9])
Combinatorial Representation Theory
We attempt to survey the field of combinatorial representation theory,
describe the main results and main questions and give an update of its current
status. We give a personal viewpoint on the field, while remaining aware that
there is much important and beautiful work that we have not been able to
mention
Traces in Complex Hyperbolic Triangle Groups
We present several formulas for the traces of elements in complex hyperbolic
triangle groups generated by complex reflections.
The space of such groups of fixed signature is of real dimension one. We
parameterise this space by a real invariant alpha of triangles in the complex
hyperbolic plane. The main result of the paper is a formula, which expresses
the trace of an element of the group as a Laurent polynomial in exp(i alpha)
with coefficients independent of alpha and computable using a certain
combinatorial winding number. We also give a recursion formula for these
Laurent polynomials and generalise the trace formulas for the groups generated
by complex mu-reflections.
We apply these formulas to prove some discreteness and some non-discreteness
results for complex hyperbolic triangle groups.Comment: 22 pages, 1 figure; prop. 11 added, typos corrected; cor. 19 removed
(not correct
A combinatorial approach to discrete geometry
We present a paralell approach to discrete geometry: the first one introduces
Voronoi cell complexes from statistical tessellations in order to know the mean
scalar curvature in term of the mean number of edges of a cell. The second one
gives the restriction of a graph from a regular tessellation in order to
calculate the curvature from pure combinatorial properties of the graph.
Our proposal is based in some epistemological pressupositions: the
macroscopic continuous geometry is only a fiction, very usefull for describing
phenomena at certain sacales, but it is only an approximation to the true
geometry. In the discrete geometry one starts from a set of elements and the
relation among them without presuposing space and time as a background.Comment: LaTeX, 5 pages with 3 figures. To appear in the Proceedings of the
XXVIII Spanish Relativity Meeting (ERE2005), 6-10 September 2005, Oviedo,
Spai
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
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