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

Abstract

A central problem in algebraic complexity, posed by J. Edmonds [16] , asks to decide if the span of a given l -tuple W = ( W 1 , … , W l ) of N × N complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing W 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 ( C N × N ) l of weight ( 1 , − 1 ) that does not vanish at W . In other words, Edmonds' problem is asking to decide if the weight ( 1 , − 1 ) belongs to the orbit semigroup of W . Let Q be an arbitrary acyclic quiver and W a representation of Q . We study the membership problem for the orbit semi-group of W by focusing on the so-called W -saturated weights. We first show that for any given W -saturated weight σ , checking if σ belongs to the orbit semigroup of W can be done in deterministic polynomial time. Next, let ( Q , R ) be an acyclic bound quiver with bound quiver algebra A = K Q / 〈 R 〉 and assume that W satisfies the relations in R . We show that if A / Ann A ( W ) is a tame algebra then any weight σ in the weight semigroup of W is W -saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.