
Journal of Lie Theory 29 (2019), No. 2, 559599 Copyright Heldermann Verlag 2019 Reduced and Nonreduced Presentations of Weyl Group Elements Sven Balnojan Lehrstuhl für Mathematik VI, Universität Mannheim, 68131 Mannheim, Germany svenbalnojan@gmail.com Claus Hertling Lehrstuhl für Mathematik VI, Universität Mannheim, 68131 Mannheim, Germany hertling@math.unimannheim.de This paper is a sequel to work of E. B. Dynkin [Semisimple subalgebras of semisimple Lie algebras, Translations of the AMS (2) 6 (1957) 111244] on subroot lattices of root lattices and to work of R. W. Carter [Conjugacy classes in the Weyl group, Comp. Math. 25 (1972) 159] on presentations of Weyl group elements as products of reflections. The quotients L/L_{1} are calculated for all irreducible root lattices L and all subroot lattices L_{1}. The reduced (i.e. those with minimal number of reflections) presentations of Weyl group elements as products of arbitrary reflections are classified. Also nonreduced presentations are studied. QuasiCoxeter elements and strict quasiCoxeter elements are defined and classified. An application to extended affine root lattices is given. A side result is that any set of roots which generates the root lattice contains a Zbasis of the root lattice. Keywords: Root system, subroot lattice, reduced presentation, quasiCoxeter element, extended affine root system. MSC: 17B22, 20F55 [ Fulltextpdf (304 KB)] for subscribers only. 