## Discrete complex reflection groups

Last update: 12 May 2014

## Notes and References

This is an excerpt of the lecture notes Discrete complex reflection groups by V.L. Popov. Lectures delivered at the Mathematical Institute, Rijksuniversiteit Utrecht, October 1980.

 1. Notation and formulation of the problem. 1.1. Notation. 1.2. Motions and reflections. 1.3. Main problem. 1.4. Irreducibility. 1.5. What is already known: $k=ℝ\text{.}$ 1.6. What is already known: $k=ℂ\text{.}$ 2. Formulation of the results. 2.1. Complexifications and real forms. 2.2. Classification of infinite irreducible complex noncrystallographic $r\text{-groups:}$ 2.3. Ingredients of the description. 2.4. Cohomology. 2.5. Description of the group of linear parts: the result. 2.6. The list of irreducible infinite crystallographic complex groups. 2.7. Equivalence. 2.8. The structure of an extension of $\text{Tran} W$ by $\text{Lin} W\text{.}$ 2.9. The rings and fields of definition of $\text{Lin} W\text{.}$ 2.10. Further remarks. 3. Several auxiliary results and the classification of irreducible infinite noncrystallographic complex $r\text{-groups.}$ 3.1. The subgroups of translations. 3.2. Some auxiliary results. 3.3. Semidirect products. 3.4. Classification of the irreducible infinite complex noncrystallographic $r\text{-groups.}$ 4. Invariant lattices 4.1. Root lattices. 4.2. The lattices with a fixed root lattice. 4.3. Further remarks on lattices. 4.4. Properties of the operator $S\text{.}$ 4.5. Root lattices and infinite $r\text{-groups.}$ 4.6. Description of the groups of linear parts: proof. 4.7. Description of root lattices. 4.8. Invariant lattices in the case $s=n+1\text{.}$ 5. The structure of $r\text{-groups}$ in the case $s=n+1\text{.}$ 5.1. The cocycle $c\text{.}$ 5.2. Example. References.