Subdivisions and local ℎ-vectors

Stanley, Richard P.

doi:10.1090/S0894-0347-1992-1157293-9

Subdivisions and local $h$-vectors
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by Richard P. Stanley

J. Amer. Math. Soc. 5 (1992), 805-851

DOI: https://doi.org/10.1090/S0894-0347-1992-1157293-9

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Abstract:

In Part I a general theory of $f$-vectors of simplicial subdivisions (or triangulations) of simplicial complexes is developed, based on the concept of local $h$-vector. As an application, we prove that the $h$-vector of a Cohen-Macaulay complex increases under “quasi-geometric” subdivision, thus establishing a special case of a conjecture of Kalai and this author. Techniques include commutative algebra, homological algebra, and the intersection homology of toric varieties. In Part II we extend the work of Part I to more general situations. First a formal generalization of subdivision is given based on incidence algebras. Special cases are then developed, in particular one based on subdivisions of Eulerian posets and involving generalized $h$-vectors. Other cases deal with Kazhdan-Lusztig polynomials, Ehrhart polynomials, and a $q$-analogue of Eulerian posets. Many applications and examples are given throughout.

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