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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Subdivisions and local $ h$-vectors


Author: Richard P. Stanley
Journal: J. Amer. Math. Soc. 5 (1992), 805-851
MSC: Primary 52B20; Secondary 05E99, 06A07, 13D40, 55U10
MathSciNet review: 1157293
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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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Additional Information

DOI: http://dx.doi.org/10.1090/S0894-0347-1992-1157293-9
PII: S 0894-0347(1992)1157293-9
Article copyright: © Copyright 1992 American Mathematical Society