Geodesic
Orbits of subsystem stabilisers
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 98-124 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\Phi$ be a reduced irreducible root system. We consider pairs $(S,X(S))$, where $S$ is a closed set of roots, $X(S)$ is its stabiliser in the Weyl group $W(\Phi)$. We are interested in such pairs maximal with respеct to the following order: $(S_1,X(S_1))\le (S_2,X(S_2))$ if $S_1\subseteq S_2$ and $X(S_1)\le X(S_2)$. Main theorem asserts that if $\Delta$ is a root subsystem such that $(\Delta,X(\Delta))$ is maximal with respect to the above order, then $X(\Delta)$ acts transitively both on the long and short roots in $\Phi\setminus\Delta$. This result is a broad generalisation of the transitivity of the Weyl group on roots of given length. 
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N. A. Vavilov; N. P. Kharchev. Orbits of subsystem stabilisers. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 14, Tome 338 (2006), pp. 98-124. http://geodesic.mathdoc.fr/item/ZNSL_2006_338_a2/

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