Division algebras and transitivity of group actions on buildings

Details Division algebras and transitivity of group actions on buildings Division algebras and transitivity of group actions on buildings

2011-05-10

arxiv.org/abs/1105.1965v1

Mathematics

Group Theory

15 pages

Scientific paper

Let D be a division algebra with center F and degree d>2. Let K|F be any splitting field. We analyze the action of D^* and SL_1(D) on the spherical and affine buildings that may be associated to GL_d(K) and SL_d(K). Specifically we inspect the failure of this action to be weakly transitive (and in particular strongly transitive), and find that in most cases this failure is quite extreme. In the affine case we can easily construct examples where the action is nonetheless Weyl transitive. Group-theoretically this yields examples of Tits subgroups that do not come from any BN-pair. We also prove some related results regarding cyclotomic subfields of division algebras over F=Q, or other global fields F with similar properties. Namely, for d>2, D cannot contain a non-central primitive 2d_{th} root of unity, and if d is odd, D cannot contain a non-central primitive d_{th} root of unity. Lastly we further inspect the action of D^* and SL_1(D) on the fundamental apartment of the relevant buildings, and find that a complete description of the action is often possible.

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