Geodesic
On the sum of roots of a closed set
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 9, Tome 289 (2002), pp. 277-286

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The invariant of closed set which is equal to the sum of roots is introduced. For a closed subset in the root system we describe symmetric and special parts, and its stabilizer in the Weyl group by means of the sum of roots. We define associated bases with respect to which the closed set has the simplest form. All possible values of root sum are determined. 
@article{ZNSL_2002_289_a15,
     author = {E. A. Sopkina},
     title = {On the sum of roots of a closed set},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {277--286},
     publisher = {mathdoc},
     volume = {289},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2002_289_a15/}
}
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E. A. Sopkina. On the sum of roots of a closed set. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 9, Tome 289 (2002), pp. 277-286. http://geodesic.mathdoc.fr/item/ZNSL_2002_289_a15/