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Subdivisions and local $$h$$-vectors. (English) Zbl 0768.05100
Summary: 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.

##### MSC:
 05E99 Algebraic combinatorics 06A07 Combinatorics of partially ordered sets 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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