Orbits of subsystem stabilizers
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Let Φ be a reduced irreducible root system. We consider pairs (S, X (S)), where S is a closed set of roots, X(S) is its stabilizer in the Weyl group W(Φ). We are interested in such pairs maximal with respect to the following order: (S1, X (S1)) ≤ (S2, X (S2)) if S1 ⊆ S2 and X(S1) ≤ X(S2). The main theorem asserts that if Δ is a root subsystem such that (Δ, X (Δ)) is maximal with respect to the above order, then X (Δ) acts transitively both on the long and short roots in Φ \ Δ. This result is a wide generalization of the transitivity of the Weyl group on roots of a given length. Bibliography: 23 titles.
KeywordsRoot System Conjugacy Class Weyl Group Wreath Product Fundamental System
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