Conditions on a finite number of orbits for \(A_r\)-type quivers. (Q1882999)

Let \(Q\) be a quiver of type \(A_r\). The objective of the paper is the variety \(R_d\) of \(\mathbb{C}\)-representations of \(Q\) of fixed dimension vector \(d=(d_1,\dots,d_r)\), equipped with the natural action of a group \(G\) which has the form \(\prod_{i=1}^rG_i\), where each \(G_i\) is either \(\text{Gl}(d_i,\mathbb{C})\) or \(\text{Sl}(d_i,\mathbb{C})\). \(\mathbb{C}\) denotes the field of complex numbers. It is well known that when \(G_i=\text{Gl}(d_i,\mathbb{C})\) for every \(i\) then this action has finitely many orbits. One of the main results of the paper characterizes precisely those dimension vectors and groups \(G\) of the above form which admit only a finite number of orbits in \(R_d\). The condition does not depend on the orientation of \(Q\). Moreover, an explicit description of all orbits in \(R_d\) is given in case their number is finite. (Also submitted to MR.)