#### Document Type

Article

#### Date

6-30-2006

#### Embargo Period

11-14-2011

#### Disciplines

Mathematics

#### Description/Abstract

This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)-admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)-admissible sequence annihilating the module. A (+)-admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following application that strengthens known results of Howlett and Fomin-Zelevinsky. For any fixed Coxeter element of the Weyl group associated to an indecomposable symmetric generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words.

#### Recommended Citation

Kleiner, Mark and Pelley, Allen, "Admissible Sequences, Preprojective Modules, and Reduced Words in the Weyl Group of a Quiver" (2006). *Mathematics Faculty Scholarship.* Paper 76.

http://surface.syr.edu/mat/76

#### 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/0607001