Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups

Details Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups

2006-11-04

arxiv.org/abs/math/0611106v2

Mathematics

Combinatorics

Final version -- to appear in Memoirs of the American Mathematical Society. Many small improvements in exposition, especially

Scientific paper

This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset $NC^{(k)}(W)$ for each finite Coxeter group $W$ and for each positive integer $k$. When $k=1$, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When $W$ is the symmetric group, we obtain the poset of classical $k$-divisible noncrossing partitions, first studied by Edelman. 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 algebraic noncrossing partitions $NC(W)$. This involves studying a finite Coxeter group $W$ with respect to its generating set $T$ of {\em all} reflections, instead of the usual Coxeter generating set $S$. This is the first time that this material has appeared in one place. Finally, it turns out that our poset $NC^{(k)}(W)$ shares many enumerative features in common with the ``generalized nonnesting partitions'' of Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In particular, there is a generalized ``Fuss-Catalan number'', with a nice closed formula in terms of the invariant degrees of $W$, that plays an important role in each case. We give a basic introduction to these topics, and we describe several conjectures relating these three families of ``Fuss-Catalan objects''.

Affiliated with

Also associated with

No associations

Rating

Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-594919