[Paper Review] Reflection Groups. A Contribution to the Handbook of Algebra

Meinolf Geck, Gunter Malle|ArXiv.org|Nov 2, 2003

Finite Group Theory Research148 references3 citations

TL;DR

This chapter provides a comprehensive overview of finite reflection groups over real and complex vector spaces, emphasizing their classification, Coxeter-type presentations, topological and geometric structures, and connections to Hecke algebras and representation theory. The key contribution is a unified treatment of the theory, including the classification of complex reflection groups, the role of invariants in polynomial rings, and the emergence of p-adic and p-compact groups via lattice invariants and homotopy-theoretic constructions.

ABSTRACT

This is a survey article on the theory of finite complex reflection groups. No proofs are given but numerous references are included.

Motivation & Objective

• To provide a systematic overview of finite reflection groups over real and complex vector spaces, integrating classical and modern perspectives.
• To clarify the classification of irreducible finite reflection groups, including infinite families and exceptional cases.
• To explore the interplay between combinatorial presentations (Coxeter-type), geometric invariants, and topological structures such as braid groups.
• To examine the role of reflection groups in representation theory, especially through Hecke algebras and the abelian defect group conjecture.
• To investigate the behavior of invariants over p-adic integers and their connection to p-compact groups in homotopy theory.

Proposed method

• Use of the Shephard–Todd classification to organize irreducible complex reflection groups into infinite families and exceptional cases.
• Application of invariant theory to show that the ring of invariants under a finite reflection group is a polynomial ring, especially over complex or p-adic fields.
• Employment of Coxeter–Dynkin diagrams to encode the presentation of finite real reflection groups via generators and relations.
• Construction of braid groups as fundamental groups of complement spaces of hyperplane arrangements associated with reflection groups.
• Utilization of p-adic lattices and torsion conditions to determine when invariants remain polynomial over Z_p, especially in the context of Weyl groups.
• Application of homotopy-theoretic methods to link p-adic reflection groups to p-compact groups via Weyl group data and loop space homology.

Experimental results

Research questions

• RQ1 Which finite reflection groups over C or R admit a polynomial ring of invariants, and what characterizes such groups?
• RQ2 How do the presentations of finite reflection groups—especially Coxeter-type presentations—facilitate combinatorial and algebraic analysis?
• RQ3 What is the relationship between the topology of hyperplane arrangements and the structure of associated braid groups?
• RQ4 Under what conditions on a prime p do the invariants of a reflection group over the p-adic integers remain polynomial?
• RQ5 How do p-adic reflection groups arise as Weyl groups of p-compact groups, and what determines their classification?

Key findings

• The ring of invariants of a finite reflection group acting on a complex vector space is always a polynomial ring, a result known as the Shephard–Todd theorem.
• Finite real reflection groups are precisely the Coxeter groups, and their presentations are encoded by Dynkin diagrams with specific edge labels.
• The invariants of Weyl groups on the weight lattice over Z_p are polynomial if all torsion primes of W are invertible in R, as shown by Demazure.
• For the symmetric group S₃, the invariants over the p-adic integers are not polynomial, even though they are polynomial over Q_p and F_p.
• All p-adic reflection groups classified by Clark–Ewing arise as Weyl groups of p-compact groups, and for p > 2, the Weyl group data over Z_p classify connected p-compact groups up to isomorphism.
• The classification of p-adic reflection groups over Q_p is complete and depends on congruence conditions on p modulo the group’s parameters, as detailed in Table 6.