Semi–invariants of Quivers for Arbitrary Dimension Vectors

The representations of dimension vector α of the quiver Q can be parametrised by a vector space R(Q,α) on which an algebraic group Gl(α) acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi–invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate to a suitable map between projective representations a semi–invariant polynomial function that describes when this map is inverted on the representation and we show that these semi–invariant polynomial functions form a spanning set of all semi–invariant polynomial functions in characteristic 0. If the quiver has no oriented cycles, we may replace consideration of inverting maps between projective representations by consideration of representations that are left perpendicular to some representation of dimension vector α. These left perpendicular representations are just the cokernels of the maps between projective representations that we consider. 1. Notation and Introduction In the sequel k will be an algebraically closed field. For our main result, Theorem 2.3 below, k will have characteristic zero. Let Q be a quiver with finite vertex set V , finite arrow set A and two functions i, t : A → V where for an arrow a we shall usually write ia in place of i(a), the initial vertex, and ta = t(a), the terminal vertex. A representation, R, of the quiver Q associates a k-vector space R(v) to each vertex v of the quiver and a linear map R(a) : R(ia) → R(ta) to each arrow a. A homomorphism φ of representations from R to S is given by a collection of linear maps for each vertex φ(v) : R(v) → S(v) such that for each arrow a, R(a)φ(ta) = φ(ia)S(v). The category of representations of the quiver Rep(Q) is an abelian category as is the full subcategory of finite dimensional representations. We shall usually be interested in finite dimensional representations in which case each representation has a dimension vector dimR, which is a function from the set of vertices V to the natural numbers N defined by (dimR)(v) = dimR(v). Now let α be a dimension vector forQ. The representations of dimension vector α are parametrised by the vector space R(Q,α) = ×a∈A k where k is the vector space of m by n matrices over k (and k and k are shorthand for k and k respectively). Given a point p ∈ R(Q,α), we denote the corresponding representation by Rp. The isomorphism classes of representations of Date: February 1, 2008. 1991 Mathematics Subject Classification. Primary 13A50.