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Abstract:
We discuss connections between the preprojective representations of
a quiver, the (+)-
admissible sequences of vertices, and the Weyl group. To each preprojective
representation
corresponds 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 and Fomin-
Zelevinsky. The talk is
based on joint work with Helene R. Tyler and with Allen Pelley.
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