David B. Wales

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.

Designs derived from permutation groups

Abstract Let G be a transitive permutation group on a set Ω of v points {1, 2, …, v}. Let H be an intransitive subgroup of G and let Δ a set of k points where Δ consists of complete orbits of H. Then the images Δx of Δ under permutations x of Δ have been shown by the first author to be a partially balanced block design D with G as a group of automorphisms. Under certain circumstances D is a balanced incomplete block design. Here a representation of the simple group PSL3(4) of order 20,160 on 56 letters leads to a new symmetric block design with parameters v=56, k-11, λ=2. A representation of the simple group of order 25,920 as U4(4) on 45 isotropic points gives a symmetric design with v=45, k=12, λ=3. One representation of U4(4) on 40 points, gives the design of planes in PG(3, 3) and exhibits the isomorphism of this group to the symplectic group S4(3).

Finding the radical of an algebra of linear transformations

We present a method that reduces the problem of computing the radical of a matrix algebra over an arbitrary field to solving systems of semilinear equations. The complexity of the algorithm, measured in the number of arithmetic operations and the total number of the coefficients passed to an oracle for solving semilinear equations, is polynomial. As an application of the technique we present a simple test for isomorphism of semisimple modules.

Computing the discriminants of Brauer’s centralizer algebras

This paper discusses a computational problem arising in the study of the structure theory of Brauers orthogonal and symplectic centralizer algebras. The problem is to compute the ranks of certain combinatorially defined matrices Zm k(x) (these matrices are presented in §2). This computation is difficult because the sizes of the matrices Zm k(x) are enormous even for small values of m and k . However, there is a great deal of symmetry amongst the entries of the matrices. In this paper we show how to design algorithms that take full advantage of this symmetry, using the representation theory of the symmetric groups. We also present data collected using these algorithms and a number of conjectures about the centralizer algebras.

Finite subgroups of G2(C)

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A global code invariant under the Higman—Sims group

In this paper we consider certain doubly transitive permutation groups which have monomial representations which are reducible. These groups have been studied by Hale and Shult 14) and by Taylor [ I7 1. The stabilizer H of a vertex has a subgroup L of index two for which elements of L are not conjugate in G to elements of H-L. The monomial representations have two invariant subspaces which can be reduced modulo any prime to form codes over finite fields. The invariant subspaces form global codes. This approach is motivated by the work of Ward 1 181 who considered the groups RSLf2, q) and Sp(2n, 4). The codes for PSL(2, q) give the classical extended quadratic residue codes. In Section 2, we construct the monomial representations and determine the two invariant subspaces. Taylor constructed 2-graphs from the centralizer algebra of these representations. The invariant subspaces are spanned by vectors whose entries are parameters in these Z&graphs. We illustrate the construction in the remaining sections with the doubiv transitive permutation representation on 176 vertices of the IIigman-Sims group. This permutation representation was first discovered by Graham Higman. The stabilizer of a vertex is a group H isomorphic to PTU(3, 5). It

Linear groups of degree n containing an involution with two eigenvalues −1, II

In [12], quasiprimitive linear groups G containing a matrix with exactly two eigenvalues ~ 1, the rest 1 were shown to have a certain property. Either G was one of a specific list of groups or the product of any two of these matrices with exactly two eigenvalues 1 had order I, 2, 3, 4, or 5. If the order was 4 the square was either in a normal 2-group of G or it itself had exactly two eigenvalues -1. This means, using [I, 201, that the group generated b!these matrices with two eigenvalues -1 has known composition factors. In this paper these groups are listed explicitlyin the following theorem.

THE MULTIPLIERS OF THE SIMPLE GROUPS OF ORDER 604,800 AND 50,232,960,

Abstract : The groups of the title were first characterized by Janko in terms of the centralizer of a central involution. If there are two classes of involutions, the group is the Hall-Janko group of order 604,800 = (2 to the 7th power) x (3 cubed) x (5 squared) x 7. It was first constructed by M. Hall, Jr. and we denote it by J sub 2. Otherwise there is only one class of involutions and the group is of order 50,232,960 = (2 to the 7th power) x (3 to the 5th power) x 5 x 17 x 19. This group was first constructed by G. Higman and J. McKay. We denote it by J sub 3. The main result is that the multiplier of J sub 2 has order 2 and that of J sub 3 has order 3. A consequence is that J sub 3 has a projective (complex) representation of degree 18. (Author)

Finite Subgroups of F_4(C) and E_6(C)

The isomorphism types of Lie primitive finite subgroups of the complex Lie groups E_6(C) and F_4(C) are determined. Some additional information is provided, such as the characters of these finite subgroups on some small-dimensional modules for the Lie groups.

Linear groups containing an involution with two eigenvalues—1

The main theorem of this paper describes quasiprimitive linear groups G which contain a matrix with two eigenvalues 1 and the remaining eigenvalues 1. This is a special case of a linear group containing a unimodular matrix with a trivial eigenspace of codimension 2. If a linear group contains a unimodular matrix with trivial eigenspace of codimension 2 other than this, the group is known by [l], [12], or [8], as is described in [8]. In a later paper [9], we treat linear groups containing a matrix with any eigenspace of codimension 2. Of course, there we refer to this work. Linear groups containing a matrix with eigenspace of codimension 1 were determined in [14] in 1914. We prove the following theorem.