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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.

Attributes	Values
rdfs:label	H-vector (en)
rdfs:comment	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. (en)
dcterms:subject	Algebraic combinatorics Polyhedral combinatorics
Wikipage page ID	11503563 (xsd:integer)
Wikipage revision ID	1071126878 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic combinatorics Algebraic combinatorics Peter McMullen Richard P. Stanley Dehn–Sommerville equations Invariant (mathematics) Ranked poset Louis Billera Ehrhart polynomial Graded algebra Simplicial sphere Krull dimension G-conjecture G-theorem Poincaré duality Abstract simplicial complex Polyhedral combinatorics Pascal's triangle Graded poset Hilbert–Poincaré series Projective variety Margaret Bayer Karim Adiprasito Toric variety Inclusion–exclusion principle Octahedron Order complex Euler characteristic Eulerian poset Finitely generated algebra Simplicial polytope Stanley–Reisner ring Maximal chain Intersection cohomology
Link from a Wikipage to an external page	http://www-math.mit.edu/~rstan/ec/
sameAs	H-vector H-vector H-vector
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Reflist dbt:Lowercase
has abstract	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. (en)
gold:hypernym	Invariant
prov:wasDerivedFrom	wikipedia-en:H-vector?oldid=1071126878&ns=0
page length (characters) of wiki page	13860 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:H-vector
is Link from a Wikipage to another Wikipage of	Toric h-vector Dehn–Sommerville equations Upper bound theorem Combinatorial commutative algebra Polyhedral combinatorics Polymake Caroline Klivans Eulerian poset Cd-index Flag f-vector Flag h-vector
is Wikipage redirect of	Toric h-vector Cd-index Flag f-vector Flag h-vector
is foaf:primaryTopic of	wikipedia-en:H-vector

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