Cluster Complexes via Semi-Invariants

Details Cluster Complexes via Semi-Invariants Cluster Complexes via Semi-Invariants

2007-08-06

arxiv.org/abs/0708.0798v2

Mathematics

Representation Theory

34 pages

Scientific paper

We define and study virtual representation spaces having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the First Fundamental Theorem, the Saturation Theorem and the Canonical Decomposition Theorem. In the special case of Dynkin quivers with n vertices this gives the fundamental interrelationship between supports of the semi-invariants and the Tilting Triangulation of the (n-1)-sphere.

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