Hodge structures on posets

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 $k$.