The Hodge Structure on a Filtered Boolean Algebra |
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Authors: | Scott Kravitz |
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Affiliation: | (1) Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003, USA |
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Abstract: | Let (Bn) be the order complex of the Boolean algebra and let B(n, k) be the part of (Bn) where all chains have a gap at most k between each set. We give an action of the symmetric group Sl on the l-chains that gives B(n, k) a Hodge structure and decomposes the homology under the action of the Eulerian idempontents. The Sn 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.We then exploit the connection between chains and tabloids to give various special cases of the homology. Also an upper bound is obtained using spectral sequence methods.Finally we present some data on the homology of B(n, k). |
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Keywords: | Hodge structure Boolean algebra Euler characteristics |
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