50 research outputs found

    Toric topology

    Full text link
    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

    Full text link
    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

    Full text link
    In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group P\mathcal{P} generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product Γ—\times and a join ⋇\divideontimes of polytopes. (P,Γ—)(\mathcal{P},\times) is a commutative associative bigraded ring of polynomials, and RP=(Zβˆ…βŠ•P,⋇)\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes) is a commutative associative threegraded ring of polynomials. The ring RP\mathcal{RP} has the structure of a graded Hopf algebra. It turns out that P\mathcal{P} has a natural Hopf comodule structure over RP\mathcal{RP}. Faces operators dkd_k that send a polytope to the sum of all its (nβˆ’k)(n-k)-dimensional faces define on both rings the Hopf module structures over the universal Leibnitz-Hopf algebra Z\mathcal{Z}. This structure gives a ring homomorphism \R\to\Qs\otimes\R, where R\R is P\mathcal{P} or RP\mathcal{RP}. Composing this homomorphism with the characters Pnβ†’Ξ±nP^n\to\alpha^n of P\mathcal{P}, Pnβ†’Ξ±n+1P^n\to\alpha^{n+1} of RP\mathcal{RP}, 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 FF 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 Q\mathbb Q, and find the relations between the images, the homomorphisms and the Hopf comodule structures. For each homomorphism f,β€…β€ŠfRPf,\;f_{\mathcal{RP}}, and \F the images of two polytopes coincide if and only if they have equal flag ff-vectors. Therefore algebraic structures on the images give the information about flag ff-vectors of polytopes.Comment: 61 page

    h-vectors of Gorenstein* simplicial posets

    Get PDF
    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

    Full text link
    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
    corecore