Polytopes of high rank for the alternating groups

There is a well-known correspondence between abstract regular polytopes and string C-groups. In this paper, for each d?3, a string C-group with d generators, isomorphic to an alternating group of degree n is constructed (for some n?9), or equivalently an abstract regular d-polytope, is produced with automorphism group Alt(n). A method that extends the CPR graph of a polytope to a different CPR graph of a larger (or possibly isomorphic) polytope is used to prove that various groups are themselves string C-groups.     

Minimal covers of the prisms and antiprisms

This paper contains a classification of the regular minimal abstract polytopes that act as covers for the convex polyhedral prisms and antiprisms. It includes a detailed discussion of their topological structure, and completes the enumeration of such covers for convex uniform polyhedra. Additionally, this paper addresses related structural questions in the theory of string C-groups.     

Chiral polytopes from hyperbolic honeycombs

Abstract polytopes are partially ordered structures which generalize the notion of polyhedra in a combinatorial sense. This concept includes all of the classical regular polytopes as well as many other well-known configurations. Chiral polytopes are abstract polytopes with maximal rotational symmetry which lack reflexive symmetry. We employ hyperbolic geometry to derive toroidal abstract polytopes of type {6, 3, 4} and {6, 3, 5} which are either regular or chiral. Their rotation groups are projective linear groups over finite rings.     

Twisted Groups and Locally Toroidal Regular Polytopes

In recent years, much work has been done on the classification of abstract regular polytopes by their local and global topological type. Abstract regular polytopes are combinatorial structures which generalize the well-known classical geometric regular polytopes and tessellations. In this context, the classical theory is concerned with those which are of globally or locally spherical type. In a sequence of papers, the authors have studied the corresponding classification of abstract regular polytopes which are globally or locally toroidal. Here, this investigation of locally toroidal regular polytopes is continued, with a particular emphasis on polytopes of ranks and . For large classes of such polytopes, their groups are explicitly identified using twisting operations on quotients of Coxeter groups. In particular, this leads to new classification results which complement those obtained elsewhere. The method is also applied to describe certain regular polytopes with small facets and vertex-figures.

     

Abstract polytopes and the hirsch conjecture

Adler and Dantzig [2] show that abstract polytopes include simple polytopes as a special case. Further they extend the results of Klee and Walkup [3] to show that the Hirsch conjecture holds for the larger class of abstract polytopes that have dimension less than or equal to five.It is the purpose of this paper to further extend to abstract polytopes another result from Klee and Walkup [3] which states that the Hirsch conjecture is mathematically equivalent to three other statements. This result makes it possible to look at the Hirsch conjecture by applying the well-defined structure and theorems of abstract polytopes to any of its four equivalent statements.     

Hypergraph polytopes

We investigate a family of polytopes introduced by E.M. Feichtner, A. Postnikov and B. Sturmfels, which were named nestohedra. The vertices of these polytopes may intuitively be understood as constructions of hypergraphs. Limit cases in this family of polytopes are, on the one end, simplices, and, on the other end, permutohedra. In between, as notable members one finds associahedra and cyclohedra. The polytopes in this family are investigated here both as abstract polytopes and as realized in Euclidean spaces of all finite dimensions. The later realizations are inspired by J.D. Stasheff ?s and S. Shnider?s realizations of associahedra. In these realizations, passing from simplices to permutohedra, via associahedra, cyclohedra and other interesting polytopes, involves truncating vertices, edges and other faces. The results presented here reformulate, systematize and extend previously obtained results, and in particular those concerning polytopes based on constructions of graphs, which were introduced by M. Carr and S.L. Devadoss.     

Normal polytopes: Between discrete,continuous, and random

The first three sections of this survey represent an updated and much expanded version of the abstract of my talk at FPSAC'2010: new results are incorporated and several concrete conjectures on the interactions between the three perspectives on normal polytopes in the title are proposed. The last section outlines new challenges in general convex polytopes, motivated by the study of normal polytopes.     

Regular Polytopes of Full Rank

The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank. Similarly, that of a discrete faithful realization of a regular apeirotope is at least one fewer than the rank. Realizations which attain the minimum are said to be of full rank. The regular polytopes and apeirotopes of full rank in two and three dimensions were classified in an earlier paper. In this paper these polytopes and apeirotopes are classified in all dimensions. Moreover, it is also shown that there are no chiral polytopes of full rank.     

String C-groups with real Schur index 2

We give examples of finite string C-groups (the automorphism groups of abstract regular polytopes) that have irreducible characters of real Schur index 2. This answers a problem of Monson concerning these groups.     

Mixing chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphism group has two orbits on the flags, such that adjacent flags belong to distinct orbits. Examples of chiral polytopes have been difficult to find. A ??mixing?? construction lets us combine polytopes to build new regular and chiral polytopes. By using the chirality group of a polytope, we are able to give simple criteria for when the mix of two polytopes is chiral.     

Manifold structures on abstract regular polytopes

Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS ³, connected sums of handlesS ¹ × S ² , euclidean and spherical space forms, and other examples with non-trivial fundamental group.     

Mixing regular convex polytopes

The mixing operation for abstract polytopes gives a natural way to construct a minimal common cover of two polytopes. In this paper, we apply this construction to the regular convex polytopes, determining when the mix is again a polytope, and completely determining the structure of the mix in each case.     

Colorful polytopes and graphs

The paper investigates connections between abstract polytopes and properly edge colored graphs. Given any finite n-edge-colored n-regular graph G, we associate to G a simple abstract polytope P _G of rank n, the colorful polytope of G, with 1-skeleton isomorphic to G. We investigate the interplay between the geometric, combinatorial, or algebraic properties of the polytope P _G and the combinatorial or algebraic structure of the underlying graph G, focussing in particular on aspects of symmetry. Several such families of colorful polytopes are studied including examples derived from a Cayley graph, in particular the graphicahedra, as well as the flagadjacency polytopes and related monodromy polytopes associated with a given abstract polytope. The duals of certain families of colorful polytopes have been important in the topological study of colored triangulations and crystallization of manifolds.     

局部C线性算子群

本文定义了一种局部C群，讨论了局部C群与局部C半群的关系及一些基本性质，并建立了局部C群的生成定理．     

Three-Dimensional Classical Groups Acting on Polytopes

Let V be a three-dimensional vector space over a finite field. We show that any irreducible subgroup of GL(V) that arises as the automorphism group of an abstract regular polytope preserves a nondegenerate symmetric bilinear form on V. In particular, the only classical groups on V that arise as automorphisms of such polytopes are the orthogonal groups.     

The graph of an abstract polytope

Recently a generalization of simple convex polytopes to combinatorial entities known as abstract polytopes has been proposed. The graph of an abstract polytope of dimensiond is a regular connected graph of degreed. Given a connected regular graph of degreed, it is interesting to find out whether it is the graph of some abstract polytopeP. We obtain necessary and sufficient conditions for this, in terms of the existence of a class of simple cycles in satisfying certain properties. The main result in this paper is that if a pair of simple convex polytopes or abstract polytopes have the same two-dimensional skeleton, then they are isomorphic. Every two-dimensional face of a simple convex polytope or an abstract polytope is a simple cycle in its graph. Given the graph of a simple convex polytope or an abstract polytope and the simple cycles in this graph corresponding to all its two-dimensional faces, then we show how to construct all its remaining faces. Given a regular connected graph and a class of simple cylesD in it, we provide necessary and sufficient conditions under whichD is the class of two-dimensional faces of some abstract polytope which has as its graph.This research has been partially supported by the ISDOS Research Project at the Department of Industrial and Operations Engineering, and by the National Science Foundation under Grant No. GK-27872 with the University of Michigan.     

All Polytopes Are Quotients, and Isomorphic Polytopes Are Quotients by Conjugate Subgroups

In this paper it is shown that any (abstract) polytope is a quotient of a regular polytope by some subgroup N of the automorphism group W of , and also that isomorphic polytopes are quotients of by conjugate subgroups of W . This extends work published in 1980 by Kato, who proved these results for a restricted class of polytopes which he called ``regular'. The methods used here are more elementary, and treat the problem in a purely nongeometric manner. Received January 27, 1997, and in revised form October 1, 1997.     

Open problems on k-orbit polytopes

We present 35 open problems on combinatorial, geometric and algebraic aspects of $k$-orbit abstract polytopes. We also present a theory of rooted polytopes that has appeared implicitly in previous work but has not been formalized before.     

Essential hyperbolic Coxeter polytopes

We introduce a notion of an essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter polytopes. We determine a potentially large combinatorial class of polytopes containing, in particular, all the compact hyperbolic Coxeter polytopes of dimension at least 6 which are known to be essential, and prove that this class contains finitely many polytopes only. We also construct an effective algorithm of classifying polytopes from this class, realize it in the four-dimensional case, and formulate a conjecture on finiteness of the number of essential polytopes.