Dyer, M. J.; Lehrer, G. I. Parabolic subgroup orbits on finite root systems. (English) Zbl 1398.20048 J. Pure Appl. Algebra 222, No. 12, 3849-3857 (2018). Summary: T. Oshima’s Lemma [“A classification of subsystems of a root system”, Preprint, arXiv:0611904] describes the orbits of parabolic subgroups of irreducible finite Weyl groups on crystallographic root systems. This note generalises that result to all root systems of finite Coxeter groups, and provides a self contained proof, independent of the representation theory of semisimple complex Lie algebras. Cited in 2 Documents MSC: 20F55 Reflection and Coxeter groups (group-theoretic aspects) 17B22 Root systems 20E07 Subgroup theorems; subgroup growth Keywords:root system; finite Coxeter group PDF BibTeX XML Cite \textit{M. J. Dyer} and \textit{G. I. Lehrer}, J. Pure Appl. Algebra 222, No. 12, 3849--3857 (2018; Zbl 1398.20048) Full Text: DOI arXiv References: [1] Bourbaki, N., 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, vol. 1337, (1968), Hermann Paris · Zbl 0186.33001 [2] Carter, R. W., Conjugacy classes in the Weyl group, Compos. Math., 25, 1-59, (1972) · Zbl 0254.17005 [3] Dyer, M. J.; Lehrer, G. I., Reflection subgroups of finite and affine Weyl groups, Trans. Am. Math. Soc., 363, 11, 5971-6005, (2011) · Zbl 1243.20051 [4] Dyer, M. J.; Lehrer, G. I., Geometry of certain finite Coxeter group actions, (2017), preprint · Zbl 07035923 [5] Dyer, Matthew, Reflection subgroups of Coxeter systems, J. Algebra, 135, 1, 57-73, (1990) · Zbl 0712.20026 [6] Dyer, Matthew, Imaginary cone and reflection subgroups of Coxeter groups, (October 2012), preprint [7] Dyer, Matthew J., Embeddings of root systems. I. root systems over commutative rings, J. Algebra, 321, 11, 3226-3248, (2009) · Zbl 1181.20036 [8] Victor, G. Kac, Infinite-dimensional Lie algebras, (1990), Cambridge University Press Cambridge · Zbl 0716.17022 [9] Oda, Hiroshi; Oshima, Toshio, Minimal polynomials and annihilators of generalized Verma modules of the scalar type, J. Lie Theory, 16, 1, 155-219, (2006) · Zbl 1101.22010 [10] Oshima, Toshio, A classification of subsystems of a root system, (2006) [11] Pilkington, A.; Ferdinands, T., A note on sums of roots, Rocky Mt. J. Math., (2018), in press · Zbl 1431.17007 [12] Steinberg, Robert, Lectures on Chevalley groups, (1968), Yale University New Haven, Conn., notes prepared by John Faulkner and Robert Wilson · Zbl 1196.22001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.