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Generalized noncrossing partitions and combinatorics of Coxeter groups


About this Title

Drew Armstrong, Department of Mathematics, Cornell University, Ithaca, New York 14853

Publication: Memoirs of the American Mathematical Society
Publication Year 2009: Volume 202, Number 949
ISBNs: 978-0-8218-4490-8 (print); 978-1-4704-0563-2 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-09-00565-1
Published electronically: July 22, 2009
MathSciNet review: 2561274
Keywords: Noncrossing partition, Coxeter group, Coxeter element, Catalan number, Fuss-Catalan number, nonnesting partition, cluster complex.
MSC (2000): Primary 05-02; Secondary 05A17, 05E15, 05E18, 06A06, 20F55

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Coxeter groups and noncrossing partitions
  • Chapter 3. -Divisible noncrossing partitions
  • Chapter 4. The classical types
  • Chapter 5. Fuss-Catalan combinatorics

Abstract


This memoir is a refinement of the author's PhD thesis — written at Cornell University (2006). It is primarily a desription of new research but we have also included a substantial amount of background material. At the heart of the memoir we introduce and study a poset for each finite Coxeter group and each positive integer . When , our definition coincides with the generalized noncrossing partitions introduced by Brady and Watt in 's for Artin groups of finite type and Bessis in The dual braid monoid. When is the symmetric group, we obtain the poset of classical -divisible noncrossing partitions, first studied by Edelman in Chain enumeration and non-crossing partitions.

In general, we show that is a graded join-semilattice whose elements are counted by a generalized “Fuss-Catalan number” which has a nice closed formula in terms of the degrees of basic invariants of . We show that this poset is locally self-dual and we also compute the number of multichains in , encoded by the zeta polynomial. We show that the order complex of the poset is shellable (hence Cohen-Macaulay) and we compute its homotopy type. Finally, we show that the rank numbers of are polynomials in with nonzero rational coefficients alternating in sign. This defines a new family of polynomials (called “Fuss-Narayana”) associated to the pair . We observe some interesting properties of these polynomials.

In the case that is a classical Coxeter group of type or , we show that is isomorphic to a poset of “noncrossing” set partitions in which each block has size divisible by . This motivates our general use of the term “-divisible noncrossing partitions” for the poset . In types and we prove “rank-selection” and “type-selection” formulas refining the enumeration of multichains in . We also describe bijections relating multichains of classical noncrossing partitions to “-divisible” and “-equal” noncrossing partitions. Our main tool is the family of Kreweras complement maps.

Along the way we include a comprehensive introduction to related background material. Before defining our generalization , we develop from scratch the theory of the generalized noncrossing partitions as defined by Brady and Watt in 's for Artin groups of finite type and Bessis in The dual braid monoid. This involves studying a finite Coxeter group with respect to its generating set of all reflections, instead of the usual Coxeter generating set . This is the first time that this material has appeared together.

Finally, it turns out that our poset shares many enumerative features in common with the generalized nonnesting partitions of Athanasiadis in Generalized Catalan numbers, Weyl groups and arrangements of hyperplanes and in On a refinement of the generalized Catalan numbers for Weyl groups; and the generalized cluster complexes of Fomin and Reading in Generalized cluster complexes and Coxeter combinatorics. We give a basic introduction to these topics and we make several conjectures relating these three families of “Fuss-Catalan objects”.

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