Hodge structures on posets
Author:
Phil Hanlon
Journal:
Proc. Amer. Math. Soc. 134 (2006), 18571867
MSC (2000):
Primary 05E25
Published electronically:
February 17, 2006
MathSciNet review:
2215112
Fulltext PDF Free Access
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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 .
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Additional Information
Phil Hanlon
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 481091003
DOI:
http://dx.doi.org/10.1090/S0002993906083936
PII:
S 00029939(06)083936
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. DMS0073785
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.
