Hodge structures on posets

Author:
Phil Hanlon

Journal:
Proc. Amer. Math. Soc. **134** (2006), 1857-1867

MSC (2000):
Primary 05E25

DOI:
https://doi.org/10.1090/S0002-9939-06-08393-6

Published electronically:
February 17, 2006

MathSciNet review:
2215112

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a poset with unique minimal and maximal elements and . For each , let be the vector space spanned by -chains from to in . We define the notion of a *Hodge structure on * which consists of a local action of on , for each , such that the boundary map intertwines the actions of and according to a certain condition.

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

We consider the case where is , the poset of subsets of with cardinality divisible by is fixed, and is a multiple of . We prove a remarkable formula which relates the characters of acting on the Hodge pieces of the homologies of the to the characters of acting on the homologies of the posets of partitions with every block size divisible by .

**[1]**A. Björner, A. M. Garsia, and R. P. Stanley,*An introduction to Cohen-Macaulay partially ordered sets*, Ordered sets (Banff, Alta., 1981) NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 83, Reidel, Dordrecht-Boston, Mass., 1982, pp. 583–615. MR**661307****[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]**Murray Gerstenhaber and S. D. Schack,*A Hodge-type decomposition for commutative algebra cohomology*, J. Pure Appl. Algebra**48**(1987), no. 3, 229–247. MR**917209**, https://doi.org/10.1016/0022-4049(87)90112-5**[4]**Phil Hanlon,*The action of 𝑆_{𝑛} on the components of the Hodge decomposition of Hochschild homology*, Michigan Math. J.**37**(1990), no. 1, 105–124. MR**1042517**, https://doi.org/10.1307/mmj/1029004069**[5]**Jean-Louis Loday,*Partition eulérienne et opérations en homologie cyclique*, C. R. Acad. Sci. Paris Sér. I Math.**307**(1988), no. 7, 283–286 (French, with English summary). MR**958781****[6]**Richard P. Stanley,*Flag-symmetric and locally rank-symmetric partially ordered sets*, Electron. J. Combin.**3**(1996), no. 2, Research Paper 6, approx. 22 pp.}, issn=1077-8926, review=\MR{1392491},. The Foata Festschrift.**[7]**Richard P. Stanley,*Parking functions and noncrossing partitions*, Electron. J. Combin.**4**(1997), no. 2, Research Paper 20, approx. 14. The Wilf Festschrift (Philadelphia, PA, 1996). MR**1444167****[8]**Rodica Simion and Richard P. Stanley,*Flag-symmetry of the poset of shuffles and a local action of the symmetric group*, Discrete Math.**204**(1999), no. 1-3, 369–396. MR**1691879**, https://doi.org/10.1016/S0012-365X(98)00381-1

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

DOI:
https://doi.org/10.1090/S0002-9939-06-08393-6

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.