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}.     

Problems on polytopes, their groups, and realizations

Summary The paper gives a collection of open problems on abstract polytopes that were either presented at the \emph{Polytopes Day in Calgary} or motivated by discussions at the preceding \emph{Workshop on Convex and Abstract Polytopes\/} at the Banff International Research Station in May~2005.     

Locally toroidal polytopes and modular linear groups

When the standard representation of a crystallographic Coxeter group G (with string diagram) is reduced modulo the integer d≥2, one obtains a finite group G^d which is often the automorphism group of an abstract regular polytope. Building on earlier work in the case that d is an odd prime, here we develop methods to handle composite moduli and completely describe the corresponding modular polytopes when G is of spherical or Euclidean type. Using a modular variant of the quotient criterion, we then describe the locally toroidal polytopes provided by our construction, most of which are new.     

A new Petrie-like construction for abstract polytopes

This article introduces a new construction for polytopes, that may be seen as a generalisation of the Petrie dual to higher ranks. Some theoretical results are derived regarding when the construction can be expected to work, and the construction is applied to some special cases. In particular, the generalised Petrie duals of the hypercubes are enumerated.     

Extensions of regular polytopes with preassigned Schläfli symbol

We say that a (d+1)-polytope P is an extension of a polytope K if the facets or the vertex figures of P are isomorphic to K. The Schläfli symbol of any regular extension of a regular polytope is determined except for its first or last entry. For any regular polytope K we construct regular extensions with any even number as first entry of the Schläfli symbol. These extensions are lattices if K is a lattice. Moreover, using the so-called CPR graphs we provide a more general way of constructing extensions of polytopes.     

Collineations of V-Spaces

V-spaces are special non-planar ³-spaces in the sense of Betten. The group of collineations of a given V-space is a Lie group of dimension at least 4. We will determine all V-spaces with a large collineation group. If the V-space is generated by a function f, then dim 5 implies f(x)=c|x| ^p for some constants c>0, p>1.     

Semisymmetric graphs from polytopes

Every finite, self-dual, regular (or chiral) 4-polytope of type {3,q,3} has a trivalent 3-transitive (or 2-transitive) medial layer graph. Here, by dropping self-duality, we obtain a construction for semisymmetric trivalent graphs (which are edge- but not vertex-transitive). In particular, the Gray graph arises as the medial layer graph of a certain universal locally toroidal regular 4-polytope.     

On the Petrials of thin rank 3 geometries

The aim of this paper is to extend the concept of a Petrial to thin (regular) rank three geometries. The main result is that if the Petrial of a thin regular residually connected geometry Γ is also a thin geometry, the geometry Γ has a linear diagram. Received 12 October 1999; revised 25 March 2000.     

Vertex embeddings of regular polytopes

The question of when one regular polytope (finite, convex) embedds in the vertices of another, of the same dimension, leads to a fascinating interplay of geometry, combinatorics, and matrix theory, with further relations to number theory and algebraic topology. This mainly expository paper is an account of this subject, its history, and the principal results together with an outline of their proofs. The relationships with other branches of mathematics are also explained.     

On Normal Subgroups of Coxeter Groups Generated by Standard Parabolic Subgroups

We discuss one construction of nonstandard subgroups in the category of Coxeter groups. Two formulae for the growth series of such a subgroups are given. As an application we construct a flag simple convex polytope, whose f-polynomial has non-real roots. Partially supported by a KBN grant 2 P03A 017 25     

Maximal 2-Local Geometries for the Symmetric Groups

Abstract

The set 𝒩_max (G, T) consisting of all maximal 2-local subgroups of G = Sym(n) which contain T, a Sylow 2-subgroup of G, is investigated. In addition to determining the structure of the subgroups in 𝒩_max (G, T), the simplicial sets of maximal rank are classified.     

A New Family of Partial Geometries

A new infinite family of partial geometries with parameters s=3²ⁿ-1, t== (3⁴ⁿ-1), == (3²ⁿ-1) is constructed in the Hermitian graphs H(3²ⁿ) for n 1. For each geometry we describe its automorphisms and various substructures such as spreads, packings and subgeometries. A derivation process based on Baer nets in the associated affine planes is shown to yield a large number of non-isomorphic geometries from each member of the family. For n=1 we exhibit some of these derived geometries.     

The Multiplicity Free Permutation Representations of the Ree Groups 2 G 2(q), the Suzuki Groups 2 B 2(q), and Their Automorphism Groups

Abstract

Let G a simple group of type ² B ₂(q) or ² G ₂(q), where q is an odd power of 2 or 3, respectively. The main goal of this paper is to determine the multiplicity free permutation representations of G and A ≤ Aut(G) where A is a subgroup containing a copy of G. Let B be a Borel subgroup of G. If G = ² B ₂(q) we show that there is only one non-trivial multiplicity free permutation representation, namely the representation of G associated to the action on G/B. If G = ² G ₂(q) we show that there are exactly two such non-trivial representations, namely the representations of G associated to the action on G/B and the action on G/M, where M = UC with U the maximal unipotent subgroup of B and C the unique subgroup of index 2 in the maximal split torus of B. The multiplicity free permutation representations of A correspond to the actions on A/H where H is isomorphic to a subgroup containing B if G = ² B ₂(q), and containing M if G = ² G ₂(q). The problem of determining the multiplicity free representations of the finite simple groups is important, for example, in the classification of distance-transitive graphs.