Richard Kane

Affine Weyl groups

We can pass from the Weyl group of a crystallographic root system and form an infinite group that has more information about the root system, and yet still possesses a structure analogous to that of the Weyl group. Notably, it has a Coxeter group structure. This group is called the affine Weyl group. Affine Weyl groups have a number of uses. They will be used in Chapter 12 to analyze subroot systems of crystallographic root systems. They are even useful for understanding ordinary Weyl groups. This will be demonstrated in §11–6.

The cohomology of finite H -spaces as U ( M ) algebras. II

By an H -space ( X , μ) we will mean a topological space X having the homotopy type of a connected CW complex of finite type together with a basepoint preserving map μ: x × X → X with two sided homotopy unit. Let p be a prime and let / p be the integers reduced mod p . Given an H -space ( X , μ) then H *( X ; / p ) is a commutative associative Hopf algebra over the Steenrod algebra A *( p ) and H *( X ; / p ) is iso- morphic, as an algebra, to a tensor product [⊗ , where each algebra A t is generated by a single element a i (see Theorem 7.11 of (24)). The decomposition A i is called a Borel decomposition and the elements { a i } are called the Borel generators of the decomposition. The decomposition ⊗ A i and the resulting generators { a t } are far from unique. Many choices are possible. Since A *( p ) acts on H *( X ; /p) an obvious restriction would be to choose the Borel decomposition to be compatible with this action. We would like the / p module generated by the Borel generators and their iterated p th powers to be invariant under the action of A * ( p ). More precisely we would like H *( X ; /p) to be the enveloping algebra U ( M ) of an unstable Steenrod module M (see § 2). If H *( X ; / p ) admits such a choice then it is called a U ( M ) algebra. The fact that H *( X ; / p ) is a U ( M ) algebra has applications in homotopy theory. In particular there exist unstable Adams spectral sequences which can be used to calculate the homotopy groups of X (see (21) and (7)). However, the question of U ( M ) structures for mod p cohomology seems of most interest simply as a classification device for finite H -spaces.

Euclidean reflection groups

The goal of this chapter is to introduce Euclidean reflection groups. This will be done in two ways. First of all, examples of reflection groups, in the plane and in 3-space, are discussed in detail. Secondly, we provide, via a preliminary discussion of Weyl chambers and invariant theory, a suggestion of the beautiful structure theorems that hold for reflection groups, and that explain the interest in such groups. The dihedral and symmetric groups will receive particular attention.

Ring of invariants and eigenvalues

The results in this chapter are due to Springer [1]. The main concern of this chapter is, once again, the study of the eigenspaces of elements in pseudo-reflection groups. This section can be regarded as the beginning of the study of conjugacy classes in pseudo-reflection groups (as opposed to just the case of Euclidean reflection groups). We shall be studying the relation between different eigenspaces and, in particular, the question of when eigenspaces are related through the action of G. As we shall see, the arguments of this chapter fit into the framework of algebraic geometry. The key point is that rings of invariants have a natural interpretation, in the context of algebraic geometry, as the coordinate ring of orbit spaces.

Eigenvalues for reflection groups

The most fundamental question we can ask about the invariant theory of a pseudo-reflection group is how to calculate its degrees and exponents. In this chapter, we explain how to characterize exponents in terms of data about the eigenvalues of elements from the group. The relation given in its full generality is due to PianzolaWeiss [1]. Their result is an extension of the fundamental work of Shephard-Todd [1] and Solomon [1].

Poincaré series for the ring of covariants

The main goal of this chapter is to study the Poincare series of Euclidean reflection groups. As an application of these results, we shall demonstrate a very explicit, and effective, method of calculating the degrees and exponents of a Weyl group from its underlying crystallographic root system. This chapter relies heavily on the results of Demazure [1].

Introduction: Reflection groups and invariant theory

The concept of a reflection group is easy to explain. A reflection in Euclidean space is a linear transformation of the space that fixes a hyperplane while sending its orthogonal vectors to their negatives. A reflection group is, then, any group of transformations generated by such reflections. The purpose of this book is to study such groups and their associated invariant theory, outlining the deep and elegant theory that they possess.

Harmonics and reflection groups

In this chapter, we use harmonic elements to characterize pseudo-reflection groups over fields of characteristic zero. This characterization is then used to demonstrate that isotropy subgroups of a nonmodular pseudo-reflection group are also pseudo-reflection groups. This fact will play a crucial role in the study of regular elements in Chapter 33. All of the arguments of this section are due to Steinberg [2].

Classifications of pseudo-reflection groups

In this chapter, we sketch some of the classification results obtained for pseudo-reflection groups in the nonmodular case. The classification results given in this chapter are not used elsewhere, and are simply given as illustrations of the type of patterns that hold. We begin by sketching the Shephard-Todd classification of the finite complex pseudo-reflection groups. We then explain how this classification can be used to obtain classifications of pseudo-reflection groups over other fields. The Shephard-Todd classification is the key to all other classifications described in this chapter. We shall omit most details, and only sketch arguments.

Reflection groups and Coxeter systems

In this chapter, we shall explain how the algebraic structure of finite Euclidean reflection groups can be captured in the concept of a Coxeter system. In the next two chapters, we use this algebraic structure to classify finite reflection groups.