This material has been published in
Trans. Amer. Math. Soc. 362 (2010), 2723-2787,
the only definitive repository of the content that has been
certified and accepted after peer review. Copyright and all rights therein
are retained by the American mathematical Society.
This material may not be copied or reposted
without explicit permission.
Decomposition numbers for finite Coxeter groups
and generalised non-crossing partitions
(62 pages)
Abstract.
Given a finite irreducible Coxeter group W, a positive integer d,
and types
T1,T2,...,Td (in the sense of the classification of finite
Coxeter groups), we compute the number of
decompositions c=\si1\si2...\sid
of a Coxeter element c of
W, such that \sii is a Coxeter element in a subgroup of type
Ti in W, i=1,2,...,d, and such that the factorisation is
"minimal" in the sense that the sum of the ranks of the Ti's,
i=1,2,...,d, equals the rank of W. For the exceptional types,
these decomposition numbers had 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]. We explain that the type An decomposition
numbers had been computed by Goulden and Jackson in [Europ. J.
Combin. 13 (1992), 357-365], and that the type Bn
decomposition numbers can be extracted from results of
Bóna, Bousquet, Labelle and Leroux [Adv. Appl. Math. 24
(2000), 22-56] on map enumeration. Our formula for the type Dn
decomposition numbers is new. These results are then used to
determine, for a fixed positive integer l and fixed integers
r1<=r2<=...<=rl, the number of multi-chains
\pi1<=\pi2<=...<=\pil in Armstrong's generalised non-crossing
partitions poset, where the poset rank of \pii equals
ri, and where the "block structure" of \pi1 is prescribed.
We show that this result implies all known enumerative results on
ordinary and generalised non-crossing partitions via appropriate
summations. Surprisingly, this result is even new 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 Dn generalised non-crossing partitions poset, which, in
its turn, leads to a proof of Armstrong's F=M Conjecture in type
Dn, thus completing a computational proof of the F=M Conjecture
for all types (after the earlier case-free proof by Tzanaki
["Faces of generalized cluster
complexes and noncrossing partitions"]).
The following versions are available:
Back to Christian Krattenthaler's
home page.