Preprojective representations of valued quivers and reduced words in the Weyl group of a Kac-Moody algebra

Details Preprojective representations of valued quivers and reduced words in the Weyl group of a Kac-Moody algebra Preprojective representations of valued quivers and reduced words in the Weyl group of a Kac-Moody algebra

2006-08-24

arxiv.org/abs/math/0608612v1

Mathematics

Representation Theory

Scientific paper

This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words. The latter strengthens known results of Howlett, Fomin-Zelevinsky, and the authors.

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