Hodge structures on posets
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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 $k$.References
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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
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 1857-1867
- MSC (2000): Primary 05E25
- DOI: https://doi.org/10.1090/S0002-9939-06-08393-6
- MathSciNet review: 2215112