A complex reflection is a linear transformation which fixes each point in a hyperplane. Intuitively, it resembles the transformation an image undergoes when it is viewed through a kaleidoscope, or arrangement of mirrors. This book gives a complete classification of all groups of transformations of n-dimensional complex space which are generated by complex reflections, using the method of line systems.
In particular: irreducible groups are studied in detail, and are identified with finite linear groups; reflection subgroups of reflection groups are completely classified; the theory of eigenspaces of elements of reflection groups is discussed fully; an appendix outlines links to representation theory, topology and mathematical physics. Containing over 100 exercises ranging in difficulty from elementary to research level, this book is ideal for honours and graduate students, or for researchers in algebra, topology and mathematical physics.
1. Preliminaries
2. The groups G(m, p, n)
3. Polynomial invariants
4. Poincaré series and characterisations of reflection groups
5. Quaternions and the finite subgroups of SU2(C)
6. Finite unitary reflection groups of rank two
7. Line systems
8. The Shepherd and Todd classification
9. The orbit map, harmonic polynomials and semi-invariants
10. Covariants and related polynomial identities
11. Eigenspace theory and reflection subquotients
12. Reflection cosets and twisted invariant theory
A. Some background in commutative algebra
B. Forms over finite fields
C. Applications and further reading
D. Tables
Bibliography
Index of notation
Index.
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