Perpendicular categories, null cones and dense orbits. (Q360159)

Let \(Q\) be a Euclidean quiver and \(d\) a prehomogeneous dimension vector, i.e. there is a dense \(\mathrm{GL}(d)\)-orbit in the space \(\mathrm{rep}(Q,d)\) of \(d\)-dimensional representations of \(Q\) (over an algebraically closed base field). This implies that there exist pairwise non-isomorphic indecomposable representations \(T_1,\ldots,T_r\) with \(\mathrm{Ext}(T_i,T_j)=0\) for \(i,j=1,\ldots,r\) such that \(d=\sum\lambda_i\dim(T_i)\). Denote by \(Z_{Q,d}\) the null cone, i.e. the common zero locus in \(\mathrm{rep}(Q,d)\) of the homogeneous semi-invariants of positive degree.NEWLINENEWLINE The following problem is investigated: When does \(Z_{Q,d}\) contain a dense \(\mathrm{GL}(d)\)-orbit? The author proves that if the \(\lambda_i\) are sufficiently large, then \(Z_{Q,d}\) contains a dense \(\mathrm{GL}(d)\)-orbit if and only if the quiver of the perpendicular category of \(\bigoplus T_i\) does not have a critical crown (an \(\widetilde A_n\)-type quiver with a special orientation) as a connected component. It is also understood what the latter condition means for the position of the \(T_i\) (\(i=1,\ldots,r\)) in the Auslander-Reiten quiver associated to \(Q\).