Semi-invariants of filtered quiver representations with at most two pathways

A pathway from one vertex of a quiver to another is a reduced path. We modify the classical definition of quiver representations and we prove that semi-invariant polynomials for filtered quiver representations come from diagonal entries if and only if the quiver has at most two pathways between any two vertices. Such class of quivers includes finite $ADE$-Dynkin quivers, affine $\widetilde{A}\widetilde{D}\widetilde{E}$-Dynkin quivers, star-shaped and comet-shaped quivers. Next, we explicitly write all semi-invariant generators for filtered quiver representations for framed quivers with at most two pathways between any two vertices; this result may be used to study constructions analogous to Nakajima's affine quotient and quiver varieties, which are, in special cases, $\mathfrak{M}_0^{F^{\bullet}}(n,1) := \mu_B^{-1}(0)/\!\!/B$ and $\mathfrak{M}^{F^{\bullet}}(n,1) := \mu_B^{-1}(0)^s/B$, respectively, where $\mu_B:T^*(\mathfrak{b}\times \mathbb{C}^n)\rightarrow \mathfrak{b}^*\cong \mathfrak{gl}_n^*/\mathfrak{u}$, $B$ is the set of invertible upper triangular $n\times n$ complex matrices, $\mathfrak{b}=\lie(B)$, and $\mathfrak{u}\subseteq \mathfrak{b}$ is the biggest unipotent subalgebra.