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Generalized noncrossing partitions and combinatorics of Coxeter groups. (English) Zbl 1191.05095
Mem. Am. Math. Soc. 202, No. 949, x, 159 p. (2009).
The memoir under review is a refinement of the author’s PhD thesis, written at Cornell University in 2006. Besides the description of the author’s research, this nice text also provides a concise and unified account of the present state of the field indicated in the title. In addition, the author suggests some open problems for future research and raises several conjectures, many of which have since then been object of research by other authors.
The central object of this memoir is a generalization of the lattice of noncrossing partitions: the poset \(\text{NC}^{(k)}(W)\), defined for each finite real Coxeter group \(W\), and each extended integer \(k\in\mathbb{N}\cup\{\infty\}\), called the poset of \(k\)-divisible noncrossing partitions. When \(k=1\), this poset coincides with the poset \(\text{NC}(W)\) of generalized noncrossing partitions introduced by T. Brady and C. Watt [Geom. Dedicata 94, 225–250 (2002; Zbl 1053.20034)], and, when \(W\) is the symmetric group, it corresponds to the poset of classical \(k\)-divisible noncrossing partitions first studied by P. H. Edelman [Discrete Math. 31, 171–180 (1980; Zbl 0443.05011)].
The author presents a solid study of both the structural and the enumerative properties of \(\text{NC}^{(k)}(W)\) and, in the case that \(W\) is a classical Coxeter group of type \(A\) or \(B\), shows that \(\text{NC}^{(k)}(W)\) is isomorphic to a poset of noncrossing set partitions in which each block has size divisible by \(k\). The author also identifies many enumerative features that the poset \(\text{NC}^{(k)}(W)\) shares with the generalized nonnesting partitions of C. A. Athanasiadis [Bull. Lond. Math. Soc. 36, No. 3, 294–302 (2004; Zbl 1068.20038)], and with the generalized cluster complexes of S. Fomin and N. Reading [Int. Math. Res. Not. 2005, No. 44, 2709–2757 (2005; Zbl 1117.52017)], and presents several conjectures relating these three families of objects. These families currently include what is known as the “Fuss-Catalan objects”, whose central enumerative feature is a generalized Coxeter-Catalan number, known as Fuss-Catalan number.
Here are the chapter headings, followed by some comments:
1. Introduction.
The author sketches his motivation and gives an outline of the thesis.
2. Coxeter Groups and Noncrossing Partitions.
This chapter starts with a quick introduction to the theory of finite Coxeter systems and root systems, following the approach in [J. E. Humphreys, Reflection groups and Coxeter groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge etc.: Cambridge University Press (1990; Zbl 0725.20028)]. A combinatorial approach to classical Coxeter systems via the theory of reduced S-words and the weak order is given in Section 2.3. In the following two sections, the author describes the more recent approach to this subject using the theory of reduced T-words and the absolute order. The lattice \(\text{NC}(W)\) of noncrossing partitions corresponding to the finite Coxeter group \(W\) is defined in Section 2.6, and some of its properties are discussed. Finally, in the last section, the notions of degrees and exponents for the Coxeter group \(W\) are introduced, again following Humphreys, and the generalized Coxeter-Catalan number is defined.
3. \(k\)-Divisible Noncrossing Partitions.
This chapter is the heart of the memoir. Here, the author introduces and studies the poset \(\text{NC}^{(k)}(W)\) for each finite real Coxeter group \(W\) and each extended positive integer \(k\in\mathbb{N}\cup\{\infty\}\), called the poset of \(k\)-divisible noncrossing partitions. Structural and enumerative properties of \(\text{NC}^{(k)}(W)\) are explored in sections 3.4 through 3.7. In general, it is shown that \(\text{NC}^{(k)}(W)\) is a graded join-semilattice whose elements are counted by a generalized “Fuss-Catalan number”, which can be given by a closed formula in terms of the degrees of basic invariants of \(W\). The author also shows that this poset is locally self-dual, and, he computes the number of multichains in \(\text{NC}^{(k)}(W)\), encoded by the zeta polynomial. The order complex of the poset is shown to be shellable, and therefore Cohen-Macaulay, and, its homotopy type is computed. Finally, it is shown that the rank numbers of \(\text{NC}^{(k)}(W)\) are polynomials in \(k\) with nonzero rational coefficients alternating in sign. This defines a new family of polynomials, called “Fuss-Narayana”, associated with the pair \((W,k)\), and some properties of these polynomials are described.
4. The Classical Types.
In the first two sections of this chapter, the author introduces the classical theory of noncrossing partitions as initiated by G. Kreweras [Discrete Math. 1, 333–350 (1972; Zbl 0231.05014)]. Afterwards, he explores the results from chapter 3 in the context of classical groups, and proves some case-by-case results. In particular, combinatorial realizations of the posets \(\text{NC}^{(k)}(A_{n-1})\) and \(\text{NC}^{(k)}(B_n)\) as posets of set partitions in which each block has size divisible by \(k\) are given. Krattenthaler has recently given a similar characterization of the poset \(\text{NC}^{(k)}(D_n)\) using annular noncrossing partitions.
5. Fuss-Catalan Combinatorics.
In the final chapter of this memoir, the author describes the other two subjects that, together with the poset of \(k\)-divisible noncrossing partitions, currently comprise the Fuss-Catalan combinatorics of Coxeter groups. The first one is the theory of the generalized nonnesting partitions of Athanasiadis, and the second one is the theory of generalized cluster complexes of Fomin and Reading. Several coincidences between the three families of Fuss-Catalan objects are described, and the author makes several conjectures related to these families.

MSC:
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05A17 Combinatorial aspects of partitions of integers
05E15 Combinatorial aspects of groups and algebras (MSC2010)
05E18 Group actions on combinatorial structures
06A06 Partial orders, general
20F55 Reflection and Coxeter groups (group-theoretic aspects)
05A18 Partitions of sets
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