Normalizers of Parabolic Subgroups of Coxeter Groups
نویسنده
چکیده
We improve a bound of Borcherds on the virtual cohomological dimension of the non-reflection part of the normalizer of a parabolic subgroup of a Coxeter group. Our bound is in terms of the types of the components of the corresponding Coxeter subdiagram rather than the number of nodes. A consequence is an extension of Brink’s result that the non-reflection part of a reflection centralizer is free. Namely, the non-reflection part of the normalizer of parabolic subgroup of type D5 or Am odd is either free or has a free subgroup of index 2. Suppose Π is a Coxeter diagram, J is a subdiagram and WJ ⊆ WΠ is the corresponding inclusion of Coxeter groups. The normalizer NWΠ(WJ) has been described in detail by Borcherds [3] and BrinkHowlett [5]. Such normalizers have significant applications to working out the automorphism groups of Lorentzian lattices and K3 surfaces; see [3] and its references. NWΠ(WJ) falls into 3 pieces: WJ itself, another Coxeter group WΩ, and a group ΓΩ of diagram automorphisms of WΩ. The last two groups are called the “reflection” and “non-reflection” parts of the normalizer. Borcherds bounded the virtual cohomological dimension of ΓΩ by |J |. Our theorems 1, 3 and 4 give stronger bounds, in terms of the types of the components of J rather than the number of nodes. There are choices involved in the definition of WΩ and ΓΩ, and our bound in theorem 3 applies regardless of how these choices are made (theorem 1 is a special case). Theorem 4 improves this bound when WΩ is “maximal”. In this case, when J = D5 or Am odd, ΓΩ turns out to either be free or have an index 2 subgroup that is free. This extends Brink’s result [4] that ΓΩ is free when J = A1. The author is grateful to the Clay Mathematics Institute, the Japan Society for the Promotion of Science, and Kyoto University for their support and hospitality. We follow the notation of [3], and refer to [6] for general information about Coxeter groups. Suppose (WΠ,Π) is a Coxeter system, which is
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