Journal of Lie Theory

no. 2
Reduced and Nonreduced Presentations of Weyl Group Elements
Journal of Lie Theory, Volume 29 (2019) no. 2, pp. 559-599
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.
DOI: 10.5802/jolt.1070
Classification: 17B22, 20F55
Keywords: Root system, subroot lattice, reduced presentation, quasi-Coxeter element, extended affine root system 
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     author = {S. Balnojan and C. Hertling},
     title = {Reduced and {Nonreduced} {Presentations} of {Weyl} {Group} {Elements}},
     journal = {Journal of Lie Theory},
     pages = {559--599},
     year = {2019},
     volume = {29},
     number = {2},
     doi = {10.5802/jolt.1070},
     zbl = {1446.17023},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.1070/}
}
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S. Balnojan; C. Hertling. Reduced and Nonreduced Presentations of Weyl Group Elements. Journal of Lie Theory, Volume 29 (2019) no. 2, pp. 559-599. doi: 10.5802/jolt.1070

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