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.     

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.     

Self-Duality of Polytopes and its Relations to Vertex Enumeration and Graph Isomorphism

We study the complexity of determining whether a polytope given by its vertices or facets is combinatorially isomorphic to its polar dual. We prove that this problem is Graph Isomorphism hard, and that it is Graph Isomorphism complete if and only if Vertex Enumeration is Graph Isomorphism easy. To the best of our knowledge, this is the first problem that is not equivalent to Vertex Enumeration and whose complexity status has a non-trivial impact on the complexity of Vertex Enumeration irrespective of whether checking Self-duality turns out to be strictly harder than Graph Isomorphism or equivalent to Graph Isomorphism. The constructions employed in the proof yield a class of self-dual polytopes that are interesting on their own. In particular, this class of self-dual polytopes has the property that the facet-vertex incident matrix of the polytope is transposable if and only if the matrix is symmetrizable as well. As a consequence of this construction, we also prove that checking self-duality of a polytope, given by its facet-vertex incidence matrix, is Graph Isomorphism complete, thereby answering a question of Kaibel and Schwartz.     

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.     

Constructions for chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphismgroup has two orbits on flags, with adjacent flags lying indifferent orbits. In this paper, we describe a method for constructingfinite chiral n-polytopes, by seeking particular normal subgroupsof the orientation-preserving subgroup of an n-generator Coxetergroup (having the property that the subgroup is not normalizedby any reflection and is therefore not normal in the full Coxetergroup). This technique is used to identify the smallest examplesof chiral 3- and 4-polytopes, in both the self-dual and non-self-dualcases, and then to give the first known examples of finite chiral5-polytopes, again in both the self-dual and non-self-dual cases.     

Oriented Matroid Rigidity of Multiplices

A d -multiplex is a self-dual convex d -polytope whose quotient polytopes and faces are also multiplices. Presently, we verify that, up to 2d vertices, it has a unique underlying affine structure (oriented matroid). Received January 4, 1999, and in revised form May 13, 1999.     

Highly symmetric hypertopes

We study incidence geometries that are thin and residually connected. These geometries generalise abstract polytopes. In this generalised setting, guided by the ideas from the polytope theory, we introduce the concept of chirality, a property of orderly asymmetry occurring frequently in nature as a natural phenomenon. The main result in this paper is that automorphism groups of regular and chiral thin residually connected geometries need to be C-groups in the regular case and \({C^+}\)-groups in the chiral case.     

Eisentein Integers and Related C-groups

Using modular quotients of linear groups defined over the Eisenstein ring Z[], we construct infinite families of finite regular or chiral polytopes of types {3,3,6}, {3,6,3} and {6,3,6}.     

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.     

Correlation polytopes: Their geometry and complexity

A family of polytopes, correlation polytopes, which arise naturally in the theory of probability and propositional logic, is defined. These polytopes are tightly connected to combinatorial problems in the foundations of quantum mechanics, and to the Ising spin model. Correlation polytopes exhibit a great deal of symmetry. Exponential size symmetry groups, which leave the polytope invariant and act transitively on its vertices, are defined. Using the symmetries, a large family of facets is determined. A conjecture concerning the full facet structure of correlation polytopes is formulated (the conjecture, however, implies that NP=co-NP).Various complexity results are proved. It is shown that deciding membership in a correlation polytope is an NP-complete problem, and deciding facets is probably not even in NP. The relations between the polytope symmetries and its complexity are indicated.     

One gauge-invariant system for two-dimensional relativistic fields: Global solutions,integrability

The system originating from the theory of two-dimensional hyperbolic chiral fields and the theory of self-dual Yang-Mills fields over ?^{2 + 2} is studied. The existence of global solutions for the Cauchy problem is established. It is observed that the system is completely integrable and one method of construction of a new solution is presented.     

Quotients of polytopes and C-groups

Abstract polytopes are combinatorial and geometrical structures with a distinctive topological flavor, which resemble the convex polytopes. C-groups are generalizations of Coxeter groups and are the automorphism groups of abstract polytopes which are regular. We investigate general properties of quotients of abstract polytopes and C-groups. Supported by NSF Grant DMS-9202071.     

On permutation polytopes

In this paper we lay the foundations for the study of permutation polytopes: the convex hull of a group of permutation matrices.We clarify the relevant notions of equivalence, prove a product theorem, and discuss centrally symmetric permutation polytopes. We provide a number of combinatorial properties of (faces of) permutation polytopes. As an application, we classify ?4-dimensional permutation polytopes and the corresponding permutation groups. Classification results and further examples are made available online.We conclude with several questions suggested by a general finiteness result.     

2n Bordered constructions of self-dual codes from group rings

Self-dual codes, which are codes that are equal to their orthogonal, are a widely studied family of codes. Various techniques involving circulant matrices and matrices from group rings have been used to construct such codes. Moreover, families of rings have been used, together with a Gray map, to construct binary self-dual codes. In this paper, we introduce a new bordered construction over group rings for self-dual codes by combining many of the previously used techniques. The purpose of this is to construct self-dual codes that were missed using classical construction techniques by constructing self-dual codes with different automorphism groups. We apply the technique to codes over finite commutative Frobenius rings of characteristic 2 and several group rings and use these to construct interesting binary self-dual codes. In particular, we construct some extremal self-dual codes of length 64 and 68, constructing 30 new extremal self-dual codes of length 68.     

More on quotient polytopes

Summary. In an earlier paper, it was shown that every abstract polytope is a quotient Q = M(W)/N {\cal Q} = {\cal M}(W)/N of some regular polytope M(W) {\cal M}(W) whose automorphism group is W, by a subgroup N of W. In this paper, attention is focussed on the quotient Q {\cal Q} , and various important structures relating to polytopes are described in terms of N ', the stabilizer of a flag of the quotient under an action of W (the 'flag action'). It is pointed out how N ' may be assumed without loss of generality to equal N. The paper also shows what properties of N ' yield polytopes which are regular, section regular, chiral, locally regular, or locally universal. The aim is to make it more practical to study non-regular polytopes in terms of group theory.     

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.

     

Triality,biquaternion and vector representation of the Dirac equation

The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion. A bilinear law of composition for biquaternion is defined by means of Levi-civita symbol and Lorentzian metric only. It is proved that there exists a vector-representation of Dirac spinors. The massive Dirac equation in the vector-representation is actually self-dual. A chiral transformations for spinors is equivalent to U(1) transformations for corresponding complex vectors. The Dirac’s idea of non-integrable phases is used to study the behavior of massive term.     

On self-dual cyclic codes over finite chain rings

In this paper, we give necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite chain rings. We prove that there are no free cyclic self-dual codes over finite chain rings with odd characteristic. It is also proven that a self-dual code over a finite chain ring cannot be the lift of a binary cyclic self-dual code. The number of cyclic self-dual codes over chain rings is also investigated as an extension of the number of cyclic self-dual codes over finite fields given recently by Jia et al.