The Hodge Structure on a Filtered Boolean Algebra

Let Δ( B n ) be the order complex of the Boolean algebra and let B ( n , k ) be the part of Δ( B n ) where all chains have a gap at most k between each set. We give an action of the symmetric group S l on the l -chains that gives B ( n , k ) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The S n action on the chains induces an action on the Hodge pieces and we derive a generating function for the cycle indicator of the Hodge pieces. The Euler characteristic is given as a corollary.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46184/1/10801_2004_Article_5384412.pd