The Hodge Structure on a Filtered Boolean Algebra
Scott Kravitz
DOI: 10.1023/B:JACO.0000047293.97371.a2
Abstract
Let 
( B n) be the order complex of the Boolean algebra and let B( n, k) be the part of ( 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.
( B n) be the order complex of the Boolean algebra and let B( n, k) be the part of ( 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. 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.
Pages: 61–70
Keywords: Hodge structure; Boolean algebra; Euler characteristics
Full Text: PDF
References
1. M. Gerstenhaber and S.D. Schack, “A Hodge-type decomposition for commutative algebra cohomology,” J. Pure Appl. Algebra 48 (1987), 229-247.
2. P. Hanlon, “The action of Sn on the components of the Hodge decomposition of Hoschschild homology,” Michigan Math. J. 37 (1990), 105-124.
3. P. Hanlon, “Cyclic homology and the Macdonald conjectures,” Invent. Math. 86 (1986), 131-159.
4. P. Hanlon, “Hodge structure on posets,” Proc. AMS, to appear. KRAVITZ
5. J.L. Loday, Cyclic Homology, Springer, Berlin, 1998.
6. J.L. Loday, “Opérations sur l'homologie cyclique des algébres commutatives,” Invent. Math. 96(1) (1989), 205-230.
7. B.E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer-Verlag, NY, 2001.
8. C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, UK, 1994.
2. P. Hanlon, “The action of Sn on the components of the Hodge decomposition of Hoschschild homology,” Michigan Math. J. 37 (1990), 105-124.
3. P. Hanlon, “Cyclic homology and the Macdonald conjectures,” Invent. Math. 86 (1986), 131-159.
4. P. Hanlon, “Hodge structure on posets,” Proc. AMS, to appear. KRAVITZ
5. J.L. Loday, Cyclic Homology, Springer, Berlin, 1998.
6. J.L. Loday, “Opérations sur l'homologie cyclique des algébres commutatives,” Invent. Math. 96(1) (1989), 205-230.
7. B.E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer-Verlag, NY, 2001.
8. C. Weibel, An Introduction to Homological Algebra, Cambridge University Press, UK, 1994.
© 1992–2009 Journal of Algebraic Combinatorics
©
2012 FIZ Karlsruhe /
Zentralblatt MATH for the EMIS Electronic Edition