Each quiver \(Q=(V,A)\) defines the group \({\textrm{GL}}_Q\) , which is a product of copies \({\textrm{GL}}(n_v)\) , when v ranges through the set of vertices V. A representation of quiver is a representation of the group \({\textrm{GL}}_Q\) in the space \({\mathcal {H}}\) , which is a direct sum of matrix spaces associated with arrows \(\alpha \in A\) . The classical problem of the theory of invariants is to describe set of generators and their relations of the algebra (or field) of invariants. The problem of construction of generators for the algebra of \({\textrm{GL}}_Q\) -invariants was solved in series of papers by L. Le Bruyn, C. Procesi, and S.Donkin. An open problem is to determine their relations. The group \({\textrm{GL}}_Q\) contains the maximal unipotent subgroup \(U=U_Q\) , which is a product of \({\textrm{UT}}(n_v)\) over \(v\in V\) . The main problem is to construct the set of generators of the algebra and field of U-invariants. In the paper, we consider the equidimensional representations, i.e. \(n_v=n\) . For an arbitrary quiver, we construct the section of U-action on \({\mathcal {H}}\) and the system of free generators of the field of U-invariants. The construction of the section and system of generators depends on the choice of an arbitrary map \(\psi : V\rightarrow A\) that assign to each vertex one of the arrows incident to it.
