Relative invariants of the polynomial rings over the type \(D_ r\) quivers (Q1333224)

Let \(F\) be a representation of a \(D_r\)-type quiver with \(r\) vertices and arbitrary directed arrows. For example \[ \begin{aligned} V_1 \bullet @>f_1>> {\overset {V_2} \bullet} @>f_2>> \bullet \leftarrow \bullet \leftarrow \bullet \rightarrow \bullet \cdots @>f_{r-3}>>&{\overset V_{r-2}\bullet}\rightarrow\bullet V_{r-1} \\ & \downarrow f_r \tag{\(F\)}\\ &{\underset {V_r}\bullet}\end{aligned} \] where \(V_i\) is a finite dimensional vector space over some field \(k\) and, if \({\overset {V_i} \bullet} @>f_i>>{\overset {V_j}\bullet}\), \(f_i\) is a linear endomorphism from \(V_i\) to \(V_j\). Then one puts \(V=\bigoplus_{i\to j\text{ in }F}\Hom(V_i,V_j)\) and \(G=GL(V_1)\times\cdots\times GL(V_r)\). Then \(G\) acts on \(V\) naturally. Let \(S(V)\) be the polynomial ring over \(V\). Then the action of \(G\) on \(V\) naturally extends to the action on \(S(V)\). This paper determines the relative invariants in \(S(V)\) with respect to this action for the \(D_r\)-type quivers when the ground field \(k\) is the complex number field \(\mathbb{C}\).