Subdivisions and local ℎ-vectors

Abstract

In Part I a general theory of f f -vectors of simplicial subdivisions (or triangulations) of simplicial complexes is developed, based on the concept of local h h -vector . As an application, we prove that the h 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 h -vectors. Other cases deal with Kazhdan-Lusztig polynomials, Ehrhart polynomials, and a q q -analogue of Eulerian posets. Many applications and examples are given throughout.