Arrangements of Hyperplanes pp 289-300 | Cite as

# Orbit Types

## Abstract

In this appendix we present the tables of the orbits of *G* on *L*(*A*) and the values of *b* _{ i } ^{ X } . These results are reproduced from [175, 174]. Construction of the matrix *U*(*G*) is outlined in Section 6.4. In a complete matrix *U*(*G*), the rows index the types *T* of the orbits. We use the symbol *A* _{0} for the trivial group, the symbols in [38, p. 193] for irreducible Coxeter groups, *G*(*r,p,ℓ*) for the monomial groups, and *G* _{ m } where *m* is in [210, Table VII] for the remaining irreducible unitary reflection groups. If two orbits have type *T*, we label them *T′*, *T″*. The columns have the same indices in the same order, but these indices are omitted. The columns to the right of the matrix *U*(*G*) give the values of *b* _{ i } ^{ X } in Theorem 6.89. The *b* _{ i } ^{ X } are computed recursively using the matrix *U*(*G*) and the formula in Lemma 6.87. We list the orbits and their sizes for the exceptional groups of rank 2 in Tables C.1 and C.2. The rows index the groups. The columns index the types *T* of the orbits. This information is sufficient to construct the matrix *U*(*G*) in each case. For example, Table C.2 shows that in *G* _{15} there are two orbits *A′* _{1}, *A″* _{1} of type *A* _{1} with cardinalities 12, 6 and one orbit of type *C*(3) with cardinality 8. The matrix *U*(*G* _{15}) and the values of *b* _{ i } ^{ X } are given in Table C.3. Groups of rank ≥ 3 comprise the remaining tables.

## Keywords

Algebraic Geometry Complex Variable Cell Complex Analytic Space Algebraic Topology## Preview

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