Non-crossing partitions for classical reflection groups

Victor Reiner

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141 Scopus citations

Abstract

We introduce analogues of the lattice of non-crossing set partitions for the classical reflection groups of types B and D. The type B analogues (first considered by Montenegro in a different guise) turn out to be as well-behaved as the original non-crossing set partitions, and the type D analogues almost as well-behaved. In both cases, they are EL-labellable ranked lattices with symmetric chain decompositions (self-dual for type B), whose rank-generating functions, zeta polynomials, rank-selected chain numbers have simple closed forms.

Original languageEnglish (US)
Pages (from-to)195-222
Number of pages28
JournalDiscrete Mathematics
Volume177
Issue number1-3
DOIs
StatePublished - Dec 1 1997

Bibliographical note

Funding Information:
In the case of type B, these lattices have appeared earlier in the literature under a different guise, which we now explain. The dihedral group I2(n) of order 2n acts on the lattice of non-crossing partitions of an n element set, and for any fixed element a E I2(n), Montenegro \[19\] considered the sublattice consisting of the elements fixed by a. He computed the Mtbius function for any tr E I2(n), and computed the zeta polynomial for tra rotation. When n is even, and tr is rotation through 180 degrees, this sublattice gives exactly our type B analogue. Furthermore, although it was not explicitly stated in \[19\],f or arbitrary n and any rotation a in I2(n), the fixed sublattice 1 Work supported by NSF Mathematical Sciences Postdoctoral Research Fellowship DMS-9206371. * E-mail: reiner@math.unm.edu.

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