One-Point Suspensions and Wreath Products of Polytopes and Spheres

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here includ…

圈积与半直积的几何

The geometry of wreath and semi-direct products

本文研究了圈积与半直积在陪集几何上的扩展，证明了这些操作保持旗传递性、剩余连通性和薄性，并应用于构造几乎单群的正则多胞体。

Polyhedral Surfaces in Wedge Products

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially r…

Hirsch polytopes with exponentially long combinatorial segments

In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the upper bound~$O(n2^d)$ for the diameter of any normal pure simplicial complex of dimension~$d$ with~$n$ vertices, and the Hirsch bound $n-d$ if the complexes are, moreover, flag. In the present article, we propose a formulation of combinatorial segments which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual …

旗双重Cohen-Macaulay单纯复形的重复Lefschetz型分解及旗球面的gamma向量

Repeated Lefschetz-like decompositions for flag doubly Cohen--Macaulay simplicial complexes and gamma vectors of flag spheres

本文提出了旗双重Cohen-Macaulay单纯复形h多项式的重复Lefschetz型分解，通过迭代双重悬垂和边细分净非负集合，建立了旗球面gamma向量与其几何构造的直接联系，并利用布尔与非布尔区别衡量旗球面偏离交叉多胞体边界的程度。