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Journal of Lie Theory 31 (2021), No. 2, 413--438
Copyright Heldermann Verlag 2021

Cayley Graph Characterization of Geometric Reflections

Saeid Azam
Department of Mathematics, University of Isfahan, Isfahan, Iran
and: School of Mathematics, Institute for Research in Fundamental Sciences, Isfahan, Iran

Fatemeh Parishani
Department of Mathematics, University of Isfahan, Isfahan, Iran

We combine the covering theory of graphs introduced by Malnic, Nedela and Skoviera, the notion of a Cayley graph and the theory of reflection systems in order to obtain a new characterization of geometric reflections in the theory of extended affine Weyl groups. As an immediate byproduct, we recover that an extended affine Weyl group of nullity greater than one is not a Coxeter group, with respect to any minimal generating set.

Keywords: Extended affine Weyl groups, Cayley graphs, Coxeter groups, normalized darts, reflections.

MSC: 17B67 20F55; 05C25.

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