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Parabolic subgroup orbits on finite root systems. (English) Zbl 1398.20048
Summary: T. Oshima’s Lemma [“A classification of subsystems of a root system”, Preprint, arXiv:0611904] describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras.

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
17B22 Root systems
20E07 Subgroup theorems; subgroup growth
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