ar X iv : 0 70 8 . 07 98 v 2 [ m at h . R T ] 1 1 O ct 2 00 8 CLUSTER COMPLEXES VIA SEMI - INVARIANTS

We define and study virtual representation spaces for vectors 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.