Journal of Lie Theory 29 (2019), No. 2, 559--599
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.uni-mannheim.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) 111--244] on subroot lattices of root lattices and to work of R. W. Carter [Conjugacy classes in the Weyl group, Comp. Math. 25 (1972) 1--59] on presentations of Weyl group elements as products of reflections.
The quotients L/L1 are calculated for all irreducible root lattices L and all subroot lattices L1. 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. Quasi-Coxeter elements and strict quasi-Coxeter 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 Z-basis of the root lattice.
Keywords: Root system, subroot lattice, reduced presentation, quasi-Coxeter element, extended affine root system.
MSC: 17B22, 20F55
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