Algebraic combinatorics | Polyhedral combinatorics

H-vector

In algebraic combinatorics, the h-vector of a simplicial polytope is a fundamental invariant of the polytope which encodes the number of faces of different dimensions and allows one to express the Dehn–Sommerville equations in a particularly simple form. A characterization of the set of h-vectors of simplicial polytopes was conjectured by Peter McMullen and proved by Lou Billera and Carl W. Lee and Richard Stanley (g-theorem). The definition of h-vector applies to arbitrary abstract simplicial complexes. The g-conjecture stated that for simplicial spheres, all possible h-vectors occur already among the h-vectors of the boundaries of convex simplicial polytopes. It was proven in December 2018 by Karim Adiprasito. Stanley introduced a generalization of the h-vector, the toric h-vector, which is defined for an arbitrary ranked poset, and proved that for the class of Eulerian posets, the Dehn–Sommerville equations continue to hold. A different, more combinatorial, generalization of the h-vector that has been extensively studied is the flag h-vector of a ranked poset. For Eulerian posets, it can be more concisely expressed by means of a noncommutative polynomial in two variables called the cd-index. (Wikipedia).

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Related pages

Krull dimension | Invariant (mathematics) | Hilbert–Poincaré series | Eulerian poset | Abstract simplicial complex | Simplicial polytope | Finitely generated algebra | Euler characteristic | Projective variety | Stanley–Reisner ring | Poincaré duality | Inclusion–exclusion principle | Simplicial sphere | Pascal's triangle | Ranked poset | Graded poset | Dehn–Sommerville equations | Algebraic combinatorics | Octahedron | Toric variety | Ehrhart polynomial