50 research outputs found
Toric topology
We survey some results on toric topology.Comment: English translation of the Japanese article which appeared in
"Sugaku" vol. 62 (2010), 386-41
Recent developments in algebraic combinatorics
A survey of three recent developments in algebraic combinatorics: (1) the
Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and
(3) toric h-vectors and intersection cohomology. This paper is a continuation
of "Recent progress in algebraic combinatorics" (math.CO/0010218), which dealt
with three other topics.Comment: 30 page
Polytopes, Hopf algebras and Quasi-symmetric functions
In this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product and a
join of polytopes. is a commutative
associative bigraded ring of polynomials, and is a commutative associative
threegraded ring of polynomials. The ring has the structure of a
graded Hopf algebra. It turns out that has a natural Hopf
comodule structure over . Faces operators that send a
polytope to the sum of all its -dimensional faces define on both rings
the Hopf module structures over the universal Leibnitz-Hopf algebra
. This structure gives a ring homomorphism \R\to\Qs\otimes\R,
where is or . Composing this homomorphism with
the characters of , of
, and with the counit we obtain the ring homomorphisms
f\colon\mathcal{P}\to\Qs[\alpha],
f_{\mathcal{RP}}\colon\mathcal{RP}\to\Qs[\alpha], and
\F^*:\mathcal{RP}\to\Qs, where is the Ehrenborg transformation. We
describe the images of these homomorphisms in terms of functional equations,
prove that these images are rings of polynomials over , and find the
relations between the images, the homomorphisms and the Hopf comodule
structures. For each homomorphism , and \F the images
of two polytopes coincide if and only if they have equal flag -vectors.
Therefore algebraic structures on the images give the information about flag
-vectors of polytopes.Comment: 61 page
h-vectors of Gorenstein* simplicial posets
As is well known, h-vectors of simple (or simplicial) convex polytopes are
characterized. In fact, those h-vectors must satisfy Dehn-Sommerville equations
and some other inequalities. Simple convex polytopes determine Gorenstein*
simplicial posets and h-vectors are defined for simplicial posets. It is known
that h-vectors of Gorenstein* simplicial posets must satisfy Dehn-Sommerville
equations and that every component in the h-vectors must be non-negative. In
this paper we will show that h-vectors of Gorenstein* simplicial posets must
satisfy one more subtle condition conjectured by R. Stanley and complete
characterization of those h-vectors. Our proof is purely algebraic but the idea
of the proof stems from topology.Comment: 12 page
Foliations modeling nonrational simplicial toric varieties
We establish a correspondence between simplicial fans, not necessarily
rational, and certain foliated compact complex manifolds called LVMB-manifolds.
In the rational case, Meersseman and Verjovsky have shown that the leaf space
is the usual toric variety. We compute the basic Betti numbers of the foliation
for shellable fans. When the fan is in particular polytopal, we prove that the
basic cohomology of the foliation is generated in degree two. We give evidence
that the rich interplay between convex and algebraic geometries embodied by
toric varieties carries over to our nonrational construction. In fact, our
approach unifies rational and nonrational cases.Comment: 24 pages, 4 figures, expository changes, references updated. Link to
the journal http://j.mp/BatZaf; Int. Math. Res. Not. 2015 (Published online
February 24, 2015