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Hodge structures on posets |
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| Authors: | Phil Hanlon |
| Affiliation: | Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003 |
| 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 We consider the case where |
| Keywords: | |
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has a Hodge structure, then the families of Eulerian idempotents intertwine the boundary map, and so we get a splitting of 
into 
Hodge pieces. 
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 
.