Decomposition numbers for finite Coxeter groups and generalised non-crossing partitions

Authors:
C. Krattenthaler and T. W. Müller

Journal:
Trans. Amer. Math. Soc. **362** (2010), 2723-2787

MSC (2000):
Primary 05E15; Secondary 05A05, 05A10, 05A15, 05A18, 06A07, 20F55, 33C05

DOI:
https://doi.org/10.1090/S0002-9947-09-04981-2

Published electronically:
December 17, 2009

MathSciNet review:
2584617

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Abstract | Similar Articles | Additional Information

Abstract: Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,\dots ,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\sigma _1\sigma _2\cdots \sigma _d$ of a Coxeter element $c$ of $W$, such that $\sigma _i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,\dots ,d$, and such that the factorisation is “minimal” in the sense that the sum of the ranks of the $T_i$’s, $i=1,2,\dots ,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author in [*“Topics in Discrete Mathematics,”* M. Klazar et al. (eds.), Springer–Verlag, Berlin, New York, 2006, pp. 93–126] and [*Séminaire Lotharingien Combin.* **54** (2006), Article B54l]. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson in [*Europ. J. Combin.* **13** (1992), 357–365], albeit using a somewhat different language. We explain how to extract the type $B_n$ decomposition numbers from results of Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. **24** (2000), 22–56] on map enumeration. Our formula for the type $D_n$ decomposition numbers is new. These results are then used to determine, for a fixed positive integer $l$ and fixed integers $r_1\le r_2\le \dots \le r_l$, the number of multi-chains $\pi _1\le \pi _2\le \dots \le \pi _l$ in Armstrong’s generalised non-crossing partitions poset, where the poset rank of $\pi _i$ equals $r_i$ and where the “block structure” of $\pi _1$ is prescribed. We demonstrate that this result implies all known enumerative results on ordinary and generalised non-crossing partitions via appropriate summations. Surprisingly, this result on multi-chain enumeration is new even for the original non-crossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rank-selected chain enumeration in the type $D_n$ generalised non-crossing partitions poset, which, in turn, leads to a proof of Armstrong’s $F=M$ Conjecture in type $D_n$, thus completing a computational proof of the $F=M$ Conjecture for all types. It also allows one to address another conjecture of Armstrong on maximal intervals containing a random multi-chain in the generalised non-crossing partitions poset.

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Additional Information

**C. Krattenthaler**

Affiliation:
Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Vienna, Austria

MR Author ID:
106265

**T. W. Müller**

Affiliation:
School of Mathematical Sciences, Queen Mary & Westfield College, University of London, Mile End Road, London E1 4NS, United Kingdom

Keywords:
Root systems,
reflection groups,
Coxeter groups,
generalised non-crossing partitions,
annular non-crossing partitions,
chain enumeration,
Möbius function,
$M$-triangle,
generalised cluster complex,
face numbers,
$F$-triangle,
Chu–Vandermonde summation

Received by editor(s):
June 30, 2008

Received by editor(s) in revised form:
December 10, 2008

Published electronically:
December 17, 2009

Additional Notes:
The first author’s research was partially supported by the Austrian Science Foundation FWF, grant S9607-N13, in the framework of the National Research Network “Analytic Combinatorics and Probabilistic Number Theory”

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.