Abstract
In this extended abstract we consider the poset of weighted partitions Π _n^w, introduced by Dotsenko and Khoroshkin in their study of a certain pair of dual operads. The maximal intervals of Π _n^w provide a generalization of the lattice Π _n of partitions, which we show possesses many of the well-known properties of Π _n. In particular, we prove these intervals are EL-shellable, we compute the Möbius invariant in terms of rooted trees, we find combinatorial bases for homology and cohomology, and we give an explicit sign twisted <mathfrak>S</mathfrak>_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 Π _n^w has a nice factorization analogous to that of Π _n.
Highlights
We recall some combinatorial, topological and representation theoretic properties of the lattice Πn of partitions of the set [n] := {1, 2, . . . , n} ordered by refinement
The Mobius invariant of Πn is given by μ(Πn) = (−1)n−1(n − 1)! and the characteristic polynomial by χΠn (x) = (x − 1)(x − 2) . . . (x − n + 1)
The order complex ∆(P ) of a poset P is the simplicial complex whose faces are the chains of P ; and the proper part Pof a bounded poset P is the poset obtained by removing the minimum element0 and the maximum element1
Summary
Topological and representation theoretic properties of the lattice Πn of partitions of the set [n] := {1, 2, . . . , n} ordered by refinement. (Two prior attempts [6, 19] to establish Cohen-Macaulayness of [ˆ0, [n]i] are discussed in Remark 2.3.) The falling chains of our EL-labeling provide a generalization of the Lyndon basis for cohomology of Πn. It follows from an operad theoretic result of Vallette [20] and the Cohen-Macaulayness of each maximal interval [ˆ0, [n]i] that the following Sn-module isomorphism holds: n−1. Where Lie2(n) is the representation of Sn on the multilinear component of the free Lie algebra on n generators with two compatible brackets (defined in Section 3.2) and Hn−3((ˆ0, [n]i)) is the reduced simplicial homology of the order complex of the open interval (ˆ0, [n]i). We mention some generalizations of what is presented here
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