Complex reflection groups, braid groups, Hecke algebras.

*(English)*Zbl 0921.20046Complex reflection groups are linear groups generated by pseudo-reflections, i.e., linear transformations acting trivially on some hyperplane. Finite complex reflection groups were classified by Shephard and Todd in 1954. Finite real and complex reflection groups have many properties in common. Presentations of finite real reflection groups are usually given by Coxeter diagrams. The authors of the present paper give a complete list of diagrams for presentations of irreducible finite complex reflection groups.

Let \(V\) be a finite-dimensional complex vector space. Let \(\mathcal A\) be the set of reflecting hyperplanes of a finite subgroup \(W\) of \(\text{GL}(V)\) generated by pseudo-reflections. Let \({\mathcal M}=V-\bigcup_{H\in{\mathcal A}}H\) and let \(x_0\in{\mathcal M}\). Let \(p\colon{\mathcal M}\to{\mathcal M}/W\) be the canonical surjection. The fundamental group \(\pi_1({\mathcal M}/W,p(x_0))\) is called the braid group \(B\).

The braid diagram \({\mathcal D}_{br}\) associated to a diagram \(\mathcal D\) is the set of nodes of \(\mathcal D\) subject to all relations of \(\mathcal D\) but the orders of the nodes.

For most irreducible finite complex reflection groups the authors show the following. Let \({\mathcal N}({\mathcal D})\) be the set of nodes of the diagram \(\mathcal D\) for \(W\) given in the tables of the paper, identified with a set of pseudo-reflections in \(W\). For each \(s\in{\mathcal N}({\mathcal D})\), there exists an \(s\)-generator of the monodromy \({\mathbf s}\) in \(B\) such that the set \(\{{\mathbf s}\}_{s\in{\mathcal N}({\mathcal D})}\), together with the braid relations of \({\mathcal D}_{br}^{op}\), is a presentation of \(B\). The authors show that the center of \(B\) is infinite cyclic. They generalize the construction of classical Hecke algebras of finite Coxeter groups. Among other results they show that the zeta function of local monodromy of the discriminant of a complex reflection group \(W\) depends only on the braid diagram of \(W\).

Let \(V\) be a finite-dimensional complex vector space. Let \(\mathcal A\) be the set of reflecting hyperplanes of a finite subgroup \(W\) of \(\text{GL}(V)\) generated by pseudo-reflections. Let \({\mathcal M}=V-\bigcup_{H\in{\mathcal A}}H\) and let \(x_0\in{\mathcal M}\). Let \(p\colon{\mathcal M}\to{\mathcal M}/W\) be the canonical surjection. The fundamental group \(\pi_1({\mathcal M}/W,p(x_0))\) is called the braid group \(B\).

The braid diagram \({\mathcal D}_{br}\) associated to a diagram \(\mathcal D\) is the set of nodes of \(\mathcal D\) subject to all relations of \(\mathcal D\) but the orders of the nodes.

For most irreducible finite complex reflection groups the authors show the following. Let \({\mathcal N}({\mathcal D})\) be the set of nodes of the diagram \(\mathcal D\) for \(W\) given in the tables of the paper, identified with a set of pseudo-reflections in \(W\). For each \(s\in{\mathcal N}({\mathcal D})\), there exists an \(s\)-generator of the monodromy \({\mathbf s}\) in \(B\) such that the set \(\{{\mathbf s}\}_{s\in{\mathcal N}({\mathcal D})}\), together with the braid relations of \({\mathcal D}_{br}^{op}\), is a presentation of \(B\). The authors show that the center of \(B\) is infinite cyclic. They generalize the construction of classical Hecke algebras of finite Coxeter groups. Among other results they show that the zeta function of local monodromy of the discriminant of a complex reflection group \(W\) depends only on the braid diagram of \(W\).

Reviewer: Erich W. Ellers (Toronto)

##### MSC:

20F55 | Reflection and Coxeter groups (group-theoretic aspects) |

20F36 | Braid groups; Artin groups |

20C08 | Hecke algebras and their representations |

20F05 | Generators, relations, and presentations of groups |

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |