Abstract:We show that an ‘almost strong Lefschetz’ property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of $1$.
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- Martina Kubitzke
- Affiliation: Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, 35032 Marburg, Germany
- Email: firstname.lastname@example.org
- Eran Nevo
- Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
- MR Author ID: 762118
- Email: email@example.com
- Received by editor(s): January 25, 2008
- Received by editor(s) in revised form: March 18, 2008
- Published electronically: June 24, 2009
- Additional Notes: The first author was supported by DAAD
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
- Journal: Trans. Amer. Math. Soc. 361 (2009), 6151-6163
- MSC (2000): Primary 13F55
- DOI: https://doi.org/10.1090/S0002-9947-09-04794-1
- MathSciNet review: 2529927