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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.

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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