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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: https://doi.org/10.1090/S0065-9266-09-00565-1
Published electronically: July 22, 2009
Keywords: Noncrossing partition, Coxeter group, Coxeter element, Catalan number, Fuss-Catalan number, nonnesting partition, cluster complex.
MSC: Primary 05E15, 05E25, 05A18

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Table of Contents

Chapters

• Acknowledgements
• 1. Introduction
• 2. Coxeter Groups and Noncrossing Partitions
• 3. $k$-Divisible Noncrossing Partitions
• 4. The Classical Types
• 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 $NC^{(k)}(W)$ for each finite Coxeter group $W$ and each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady and Watt in $K(\pi , 1)$’s for Artin groups of finite type and Bessis in The dual braid monoid. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman in Chain enumeration and non-crossing partitions.

In general, we show that $NC^{(k)}(W)$ is a graded join-semilattice whose elements are counted by a generalized “Fuss-Catalan number” $\operatorname {Cat}^{(k)}(W)$ which has a nice closed formula in terms of the degrees of basic invariants of $W$. We show that this poset is locally self-dual and we also compute the number of multichains in $NC^{(k)}(W)$, 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 $NC^{(k)}(W)$ are polynomials in $k$ with nonzero rational coefficients alternating in sign. This defines a new family of polynomials (called “Fuss-Narayana”) associated to the pair $(W,k)$. We observe some interesting properties of these polynomials.

In the case that $W$ is a classical Coxeter group of type $A$ or $B$, we show that $NC^{(k)}(W)$ is isomorphic to a poset of “noncrossing” set partitions in which each block has size divisible by $k$. This motivates our general use of the term “$k$-divisible noncrossing partitions” for the poset $NC^{(k)}(W)$. In types $A$ and $B$ we prove “rank-selection” and “type-selection” formulas refining the enumeration of multichains in $NC^{(k)}(W)$. We also describe bijections relating multichains of classical noncrossing partitions to “$k$-divisible” and “$k$-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 $NC^{(k)}(W)$, we develop from scratch the theory of the generalized noncrossing partitions $NC^{(1)}(W)$ as defined by Brady and Watt in $K(\pi , 1)$’s for Artin groups of finite type and Bessis in The dual braid monoid. This involves studying a finite Coxeter group $W$ with respect to its generating set $T$ of all reflections, instead of the usual Coxeter generating set $S$. This is the first time that this material has appeared together.

Finally, it turns out that our poset $NC^{(k)}(W)$ 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”.