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On the (co)homology of the poset of weighted partitions


Authors: Rafael S. González D’León and Michelle L. Wachs
Journal: Trans. Amer. Math. Soc. 368 (2016), 6779-6818
MSC (2010): Primary 05E45; Secondary 05E15, 05E18, 06A11, 17B01
DOI: https://doi.org/10.1090/tran/6483
Published electronically: February 2, 2016
MathSciNet review: 3471077
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Abstract: We consider the poset of weighted partitions $ \Pi _n^w$, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of $ \Pi _n^w$ provide a generalization of the lattice $ \Pi _n$ of partitions, which we show possesses many of the well-known properties of $ \Pi _n$. In particular, we prove these intervals are EL-shellable, we show that the Möbius invariant of each maximal interval is given up to sign by the number of rooted trees on node set $ \{1,2,\dots ,n\}$ having a fixed number of descents, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted $ \mathfrak{S}_n$-module isomorphism from cohomology to the multilinear component of the free Lie algebra with two compatible brackets. We also show that the characteristic polynomial of $ \Pi _n^w$ has a nice factorization analogous to that of $ \Pi _n$.


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Additional Information

Rafael S. González D’León
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Email: dleon@math.miami.edu

Michelle L. Wachs
Affiliation: Department of Mathematics, University of Miami, Coral Gables, Florida 33124
Email: wachs@math.miami.edu

DOI: https://doi.org/10.1090/tran/6483
Received by editor(s): December 13, 2013
Received by editor(s) in revised form: April 13, 2014, and May 12, 2014
Published electronically: February 2, 2016
Additional Notes: The first author was supported by NSF grant DMS 1202755
The work of the second author was partially supported by a grant from the Simons Foundation (#267236) and by NSF grants DMS 0902323 and DMS 1202755.
Article copyright: © Copyright 2016 American Mathematical Society