Decomposition theorem for the cd-index of Gorenstein * posets
Richard Ehrenborg
and Kalle Karu
University of Kentucky Department of Mathematics Lexington KY 40506 USA
DOI: 10.1007/s10801-006-0055-y
Abstract
We prove a decomposition theorem for the cd-index of a Gorenstein * poset analogous to the decomposition theorem for the intersection cohomology of a toric variety. From this we settle a conjecture of Stanley that the cd-index of Gorenstein* lattices is minimized on Boolean algebras.
Pages: 225–251
Keywords: keywords cd-index; Gorenstein ^{*} posets; decomposition theorem; lattices; subdivisions
Full Text: PDF
References
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2. G. Barthel, J.-P. Brasselet, K.-H. Fieseler, and L. Kaup, “Combinatorial intersection cohomology for fans,” Tohoku Math. J. 54 (2002), 1-41.
3. M.M. Bayer and L.J. Billera, “Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets,” Invent. Math. 79(1) (1985), 143-157.
4. M.M. Bayer and A. Klapper, “A new index for polytopes,” Discrete Comput. Geom. 6 (1991), 33-47.
5. L.J. Billera and R. Ehrenborg, “Monotonicity of the cd-index for polytopes,” Math. Z. 233 (2000), 421-441.
6. T. Braden and R. MacPherson, “Intersection homology of toric varieties and a conjecture of Kalai,” Comment Math. Helv. 74 (1999), 442-455.
7. P. Bressler and V.A. Lunts, “Intersection cohomology on nonrational polytopes,” Compositio Math. 135(3) (2003), 245-278.
8. R. Ehrenborg and M. Readdy, “Coproducts and the cd-index,” J. Algebraic Combin. 8 (1998), 273-299.
9. K. Karu, “Hard Lefschetz theorem for nonrational polytopes,” Invent. Math. 157(2) (2004), 419-447.
10. K. Karu, “The cd-index of fans and posets,” Compositio Math. 142 (2006), 701-718.
11. R.P. Stanley, Enumerative Combinatorics, vol. 1, Cambridge Studies in Advanced Mathematics, 49, Cambridge University Press, Cambridge, 1997.
12. R.P. Stanley, “Flag f-vectors and the cd-index,” Math. Z. 216 (1994), 483-499.
13. R.P. Stanley, Combinatorics and Commutative Algebra, 2nd edition, Progress in Mathematics, vol. 41, Boston Basel Berlin: Birkh\ddot auser, 1996.