Cluster Complexes via Semi-Invariants

摘要

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