Decomposition numbers for finite Coxeter groups and generalised noncrossing partitions
Abstract
Given a finite irreducible Coxeter group $W$, a positive integer $d$, and types $T_1,T_2,...,T_d$ (in the sense of the classification of finite Coxeter groups), we compute the number of decompositions $c=\si_1\si_2 cdots\si_d$ of a Coxeter element $c$ of $W$, such that $\si_i$ is a Coxeter element in a subgroup of type $T_i$ in $W$, $i=1,2,...,d$, and such that the factorisation is "minimal" in the sense that the sum of the ranks of the $T_i$'s, $i=1,2,...,d$, equals the rank of $W$. For the exceptional types, these decomposition numbers have been computed by the first author. The type $A_n$ decomposition numbers have been computed by Goulden and Jackson, 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 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 $r_1\le r_2\le ...\le r_l$, the number of multichains $\pi_1\le \pi_2\le ...\le \pi_l$ in Armstrong's generalised noncrossing 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 noncrossing partitions via appropriate summations. Surprisingly, this result on multichain enumeration is new even for the original noncrossing partitions of Kreweras. Moreover, the result allows one to solve the problem of rankselected chain enumeration in the type $D_n$ generalised noncrossing partitions poset, which, in turn, leads to a proof of Armstrong's $F=M$ Conjecture in type $D_n$.
 Publication:

arXiv eprints
 Pub Date:
 April 2007
 arXiv:
 arXiv:0704.0199
 Bibcode:
 2007arXiv0704.0199K
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Group Theory;
 05E15 (Primary);
 05A05;
 05A10;
 05A15;
 05A18;
 06A07;
 20F55;
 33C05 (Secondary)
 EPrint:
 AmSLaTeX