The Lefschetz property for barycentric subdivisions of shellable complexes

Kubitzke, Martina; Nevo, Eran

doi:10.1090/S0002-9947-09-04794-1

The Lefschetz property for barycentric subdivisions of shellable complexes
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by Martina Kubitzke and Eran Nevo PDF

Trans. Amer. Math. Soc. 361 (2009), 6151-6163 Request permission

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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