A proof of the Popov conjecture for quivers.
Let \(G\) be a connected reductive algebraic group over an algebraically closed field of characteristic zero, and let \(V\) be a finite dimensional algebraic \(G\)-module. Let \(\pi: V\rightarrow V/\!\!/G\) be the corresponding categorical quotient. The Popov Conjecture [formulated in \textit{V.~L.~Popov}, Funct. Anal. Appl. 10, 242--244 (1976; Zbl 0365.20053)] claims that if \(\pi\) is equidimensional, then \(V/\!\!/G\) is an affine space. The author proves this conjecture in the case where \(V\) is the space of \(\alpha\)-dimensional representation of a quiver \(Q\) on which \(G=\text{GL}_\alpha\) acts by base-change (in this case \(V/\!\!/G\) is the space of isomorphism classes of semisimple representations of \(Q\) of dimension \(\alpha\)).
scientific article; zbMATH DE number 2066838