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A characterization of a special ordering in a root system. (English) Zbl 0799.20037

The author gives necessary and sufficient conditions for an ordering of a set of positive roots in a root system \(R\) to be associated to a reduced expression of an element of the Weyl group of \(R\) and characterizes the sets of positive roots which can be given such an ordering.

MSC:

20F55 Reflection and Coxeter groups (group-theoretic aspects)
17B20 Simple, semisimple, reductive (super)algebras
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[1] N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). · Zbl 0186.33001
[2] Howard Hiller, Geometry of Coxeter groups, Research Notes in Mathematics, vol. 54, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. · Zbl 0483.57002
[3] James E. Humphreys, Introduction to Lie algebras and representation theory, Springer-Verlag, New York-Berlin, 1972. Graduate Texts in Mathematics, Vol. 9. · Zbl 0254.17004
[4] James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. · Zbl 0725.20028
[5] Nagayoshi Iwahori, On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215 – 236 (1964). · Zbl 0135.07101
[6] George Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990), no. 1-3, 89 – 113. · Zbl 0714.17013
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