Am Arjeh Cohen

Near polygons and Fischer spaces

In this paper we exploit the relations between near polygons with lines of size 3 and Fischer spaces to classify near hexagons with quads and with lines of size three. We also construct some infinite families of near polygons.

Chapter 12 – Point-Line Spaces Related to Buildings

Publisher Summary This chapter focuses on point-line spaces related to buildings. The definition of space is very general in that any collection of nonempty, nonsingleton subsets of any set can be the set of lines of a space. A stricter class of spaces is formed by the partial linear spaces, i.e. spaces in which no two different points are on two different lines. For two distinct collinear points x, y of such a space, xy is written to denote the line containing them. A hyperplane of a space S is a proper subspace H of S, such that any line of S has at least one point in H. Thus, if S is a projective space, its hyperplanes are the usual maximal proper subspaces. A polar space is a space in which, for every point x, the subset x⊥ is either a hyperplane or the whole point set. Because of the definition, the polar spaces are gamma spaces of diameter at most 2. Singular spaces are degenerate examples of polar spaces. A polar space is said to be of rank n if its singular rank is n − 1.

On Generalized Hexagons and a Near Octagon whose Lines have Three Points

Proofs are given of the facts that any finite generalized hexagon of order (2, t) is isomorphic to the classical generalized hexagon associated with the group G2(2) or to its dual if t = 2 and that it is isomorphic to the classical generalized hexagon associated with the group 3D4(2) if t = 8. Furthermore, it is shown that any near octagon of order (2, 4; 0, 3) is isomorphic to the known one associated with the sporadic simple group HJ.

Affine polar spaces

Affine polar spaces are polar spaces from which a hyperplane (that is a proper subspace meeting every line of the space) has been removed. These spaces are of interest as they constitute quite natural examples of ‘locally polar spaces’. A characterization of affine polar spaces (of rank at least 3) is given as locally polar spaces whose planes are affine. Moreover, the affine polar spaces are fully classified in the sense that all hyperplanes of the fully classified polar spaces (of rank at least 3) are determined.

Linearity of artin groups of finite type

Recent results on the linearity of braid groups are extended in two ways. We generalize the Lawrence Krammer representation as well as Krammer’s faithfulness proof for this linear representation to Artin groups of finite type.

Some tapas of computer algebra

Grbner bases, an introduction, A.M. Cohen symbolic recipes for polynomial system solving, L. Gonzalez-Vega et al lattice reduction, F. Beukers factorization of polynomials, F. Beukers computation in associative and lie algebras, G. Ivanyos and L. Ronyai symbolic recipes for real solutions, L. Gonzalez-Vega et al Grbner bases and integer programming, G.M. Ziegler working with finite groups, H. Cuypers et al symbolic analysis of differential equations, M. van der Put Grbner bases for codes, M. de Boer and R. Pellikaan Grbner bases for decoding, M. de Boer and R. Pellikaan project 1 - automatic geometry theorem proving, T. Recio et al project 2 - the Birkhoff interpolation problem, M.-J. Gonzalez-Lopez and L. Gonzalez-Vega project 3 - the inverse kinematics problem in robotics, M.-J. Gonzalez-Lopez and L. Gonzalez-Vega project 4 - quaternion algebras, G. Ivanyos and L. Ronyai project 5 - explorations with the icosahedral group, A.M. Cohen et al project 6 - the small Mathieu groups, H. Cuypers et al project 7 - the Golay codes, M. de Boer and R. Pellikaan.

A characterization of some geometries of Lie type

For geometries associated with permutation representations of the groups of Lie type E6, E7, E8 on certain maximal parabolic subgroups (e.g. the stabilizers of root subgroups), axiom systems are given that characterize them in terms of points and lines.

Finite Quaternionic Reflection Groups

In this article the quaternionic reflection groups are classified. Such a group is defined so as to generalize the notion of reflection groups appearing in [4, 171, i.e., it is a group of linear transformations in a quaternionic vector space of dimension n < cc generated by elements that fix an (n 1)-dimensional subspace pointwise. Moreover, the notion of root system as given in 13, 41 is extended to the quaternionic case. These systems can be used to construct the groups involved and vice versa. Relevant definitions can be found in Section 1. In the following section, the imprimitive groups are determined in a relatively simple manner that resembles the complex case. The primitive groups are classified by means of their complexifications (i.e., their natural isomorphic images in the group of all invertible linear transformations in the 2n-dimensional complex vector space underlying the given quaternionic one). The primitive groups with imprimitive complexifications are dealt with in Section 3. In order to determine the remaining groups, extensive use is made of the work of’Huffman and Wales [12-15, 201 on the classification of primitive unimodular complex linear groups generated by elements that fix a subspace of codimension 2 pointwise. If the complexification of a quaternionic reflection group is primitive, it belongs to this family of groups. Section 4 is about these groups and their root systems. The main results are comprised in Theorems 2.6, 2.9, 3.6, and 4.2.

On a Theorem of Cooperstein

A theorem by Cooperstein that partially characterizes the natural geometry A n,d ( F ) of subspaces of rank d − 1 in a projective space of finite rank n over a finite field F , is somewhat strengthened and generalized to the case of an arbitrary division ring F . Moreover, this theorem is used to provide characterizations of A n ,2 ( F ) and A 5,3 ( F ) which will be of use in the characterization of other (exceptional) Lie group geometries.

Computing in groups of Lie type

We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction.