Nickolai Erokhovets
Polytopes, Hopf algebras and Quasi-symmetric functions.
Tue, 11/09/2010 - 16:01 — SomNambulAuthors: Victor M. Buchstaber, Nickolai Erokhovets
Subjects: Combinatorics
AbstractIn this paper we use the technique of Hopf algebras and quasi-symmetric
functions to study the combinatorial polytopes. Consider the free abelian group
$\mathcal{P}$ generated by all combinatorial polytopes. There are two natural
bilinear operations on this group defined by a direct product $\times $ and a
join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative
associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb
Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative
threegraded ring of polynomials.Ring of Polytopes, Quasi-symmetric functions and Fibonacci numbers.
Thu, 02/04/2010 - 16:01 — SomNambulAuthors: Victor M. Buchstaber, Nickolai Erokhovets
Subjects: Combinatorics
AbstractIn this paper we study the ring $\mathcal{P}$ of combinatorial convex
polytopes. We introduce the algebra of operators $\mathcal{D}$ generated by the
operators $d_k$ that send an $n$-dimensional polytope $P^n$ to the sum of all
its $(n-k)$-dimensional faces.Buchstaber Invariant of Simple Polytopes.
Tue, 08/25/2009 - 22:41 — SomNambulAuthors: Nickolai Erokhovets
Subjects: Algebraic Topology
AbstractIn this paper we study a new combinatorial invariant of simple polytopes,
which comes from toric topology. With each simple n-polytope P with m facets we
can associate a moment-angle complex Z_P with a canonical action of the torus
T^m. Then s(P) is the maximal dimension of a toric subgroup that acts freely on
Z_P. The problem stated by Victor M. Buchstaber is to find a simple
combinatorial description of an s-number. We describe the main properties of
s(P) and study the properties of simple n-polytopes with n+3 facets.
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