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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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