Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] A. Bjorner, A. Garsia and R. P. Stanley, ``An introduction to Cohen-Macaulay partially ordered sets", (I. Rival, ed., Reidel, Dordrecht, 1982), pp. 583-615. MR 0661307 (83i:06001)
  • [2] A. R. Calderbank, P. Hanlon and R. W. Robinson, ``Partitions into even and odd block size and some unusual characters of the symmetric groups'', Proc. London Math. J. 53 (1986), 288-320. MR0850222 (87m:20042)
  • [3] M. Gerstenhaber and S. D. Schack, ``A Hodge-type decomposition for commutative algebra cohomology'', J. Pure Appl. Algebra 48 (1987), 229-247. MR 0917209 (88k:13011)
  • [4] P. Hanlon, ``The action of $ S_n$ on the components of the Hodge decomposition of Hochschild homology'', Michigan Math. J. 37 (1990), 105-124.MR 1042517 (91g:20013)
  • [5] J.-L. Loday, ``Partitions éuleriennes et opérations en homologie cyclique'', C. R. Acad. Sci. Paris Sé. I. Math. 307 (1988), 283-286.MR 0958781 (89h:18017)
  • [6] R. P. Stanley, ``Flag-symmetric and locally rank-symmetric partially ordered sets'', Electronic J. Combinatorics 3 R6, 22 pp. MR 1392491 (98d:06006)
  • [7] R. P. Stanley, ``Parking functions and non-crossing partitions'', Electronic J. Combinatorics 4(2) R20, 14 pp. MR 1444167 (98m:05011)
  • [8] R. Simion and R. Stanley, ``Flag-symmetry of the poset of shuffles and a local action of the symmetric geometry'', Discrete Math. 204 (1999), 369-396. MR 1691879 (2000f:05090)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 05E25

Retrieve articles in all journals with MSC (2000): 05E25

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.

American Mathematical Society