Orbit Types

Abstract

In this appendix we present the tables of the orbits of G on L(A) and the values of b ^X _i . 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 ^X _i in Theorem 6.89. The b ^X _i 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 ^X _i are given in Table C.3. Groups of rank ≥ 3 comprise the remaining tables.