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Orbits of subsystem stabilizers

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Abstract

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.

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  1. St. Petersburg State University, St. Petersburg, Russia

    N. A. Vavilov & N. P. Kharchev

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  1. N. A. Vavilov
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  2. N. P. Kharchev
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 98–124.

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Vavilov, N.A., Kharchev, N.P. Orbits of subsystem stabilizers. J Math Sci 145, 4751–4764 (2007). https://doi.org/10.1007/s10958-007-0306-z

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