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Hodge structures on posets

Author: Phil Hanlon
Journal: Proc. Amer. Math. Soc. 134 (2006), 1857-1867
MSC (2000): Primary 05E25
Published electronically: February 17, 2006
MathSciNet review: 2215112
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Abstract: Let $ P$ be a poset with unique minimal and maximal elements $ \hat{0}$ and $ \hat{1}$. For each $ r$, let $ C_r(P)$ be the vector space spanned by $ r$-chains from $ \hat{0}$ to $ \hat{1}$ in $ P$. We define the notion of a Hodge structure on $ P$ which consists of a local action of $ S_{r+1}$ on $ C_r$, for each $ r$, such that the boundary map $ \partial_r: C_r\to C_{r-1}$ intertwines the actions of $ S_{r+1}$ and $ S_r$ according to a certain condition.

We show that if $ P$ has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of $ H_r(P)$ into $ r$ Hodge pieces.

We consider the case where $ P$ is $ \mathcal{B}_{n,k}$, the poset of subsets of $ \{1,2,\dots, n\}$ with cardinality divisible by $ k$ $ (k$ is fixed, and $ n$ is a multiple of $ k)$. We prove a remarkable formula which relates the characters $ \mathcal{B}_{n,k}$ of $ S_n$ acting on the Hodge pieces of the homologies of the $ \mathcal{B}_{n,k}$ to the characters of $ S_n$ acting on the homologies of the posets of partitions with every block size divisible by $ k$.

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Additional Information

Phil Hanlon
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003

Received by editor(s): December 12, 2001
Received by editor(s) in revised form: January 5, 2005
Published electronically: February 17, 2006
Additional Notes: This work was supported in part by the National Science Foundation under Grant No. DMS-0073785
Communicated by: John R. Stembridge
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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