#### Document Type

Article

#### Date

8-24-2006

#### Embargo Period

11-11-2011

#### Disciplines

Mathematics

#### Description/Abstract

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.

#### Recommended Citation

Kleiner, Mark and Pelley, Allen, "Preprojective Representations of Valued Quivers and Reduced Words in the Weyl Group of a Kac-Moody Algebra" (2006). *Mathematics Faculty Scholarship.* Paper 78.

http://surface.syr.edu/mat/78

#### Source

Harvested from arXiv.org

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 3.0 License.

## Additional Information

This manuscript is from arXiv.org for more information see http://arxiv.org/abs/math/0608612