The Procesi–Razmyslov theorem for quiver representations

We find the generators and the defining relations of any quiver representation invariant algebra. To be precise, let $R(Q,\bar k)$ be a quiver representation space with respect to the natural action of the group consisting of all isomorphisms of the quiver representations. Denote this group by $\operatorname{GL}(\bar k)$, where $\bar k$ is a dimensional vector of the quiver representation space $R(Q,\bar k)$. For example, when our quiver $Q$ has only one vertex and several loops are incidental to this vertex we have the well-known case of the adjoint action of the general linear group on the space of several $n\times n$-matrices. In the characteristic zero case Artin and Procesi described the quotient of the last variety under this action in their classic works. In the case of arbitrary infinite ground field this result can be deduced from some results by Procesi and Donkin. In a similar manner we can define the quotient of the quiver representation space $R(Q,\bar k)$ by the action of the group $\operatorname{GL}(\bar k)$. By the definition we have that $K[R(Q,\bar k)/\operatorname{GL}(\bar k)]\cong K[R(Q,\bar k)]^{\operatorname{GL}(\bar k)}$. Donkin has recently found the generators of that algebra. In this article we define a free quiver representation invariant algebra. Then we prove that the kernel of its canonical epimorphism onto $K[R(Q,\bar k)]^{\operatorname{GL}(\bar k)}$ is generated as a T-ideal by the values of the coefficients of the characteristic polynomial with sufficiently large number. This result generalizes the well-known Procesi–Razmyslov theorem about trace matrix identities. Besides, by an alternative way we can deduce Donkin's result about the generators of $K[R(Q,\bar k)]^{\operatorname{GL}(\bar k)}$.