Edmonds' problem and the membership problem for orbit semigroups of quiver representations

A central problem in algebraic complexity, posed by J. Edmonds, asks to decide if the span of a given $l$-tuple $\V=(\V_1, \ldots, \V_l)$ of $N \times N$ complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing $\V$ as a representation of the $l$-Kronecker quiver $\K_l$, Edmonds' problem can be rephrased as asking to decide if there exists a semi-invariant on the representation space $(\CC^{N\times N})^l$ of weight $(1,-1)$ that does not vanish at $\V$. In other words, Edmonds' problem is asking to decide if the weight $(1,-1)$ belongs to the orbit semigroup of $\V$. Let $Q$ be an arbitrary acyclic quiver and $\V$ a representation of $Q$. We study the membership problem for the orbit semi-group of $\V$ by focusing on the so-called $\V$-saturated weights. We first show that for any given $\V$-saturated weight $\sigma$, checking if $\sigma$ belongs to the orbit semigroup of $\V$ can be done in deterministic polynomial time. Next, let $(Q, \R)$ be an acyclic bound quiver with bound quiver algebra $A=KQ/\langle \R \rangle$ and assume that $\V$ satisfies the relations in $\R$. We show that if $A/\Ann_A(\V)$ is a tame algebra then any weight $\sigma$ in the weight semigroup of $\V$ is $\V$-saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.