Cluster Complexes via Semi-Invariants

TL;DR

This paper introduces virtual representation spaces with mixed dimensions for acyclic quivers, establishing foundational semi-invariant theorems and linking them to tilting triangulations in Dynkin cases.

Contribution

It defines virtual semi-invariants for these spaces and proves key theorems, extending classical invariant theory to a broader virtual setting.

Findings

Established the First Fundamental Theorem for virtual semi-invariants

Proved the Saturation and Canonical Decomposition Theorems in this context

Connected semi-invariant supports with tilting triangulations for Dynkin quivers

Abstract

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.