On open orbits and their complements

Abstract

1.1. We fix an algebraically closed field k and a quiver Q without oriented cycles, whose vertex set Q, we identify with { 1, . . . . n}. For a in the set Q, of arrows of Q, we denote by ta and ha the tail and the head of a, respectively. A representation X of Q consists of a family {X(i): in QO} of finite dimensional k-vector spaces and a family {X(a): X(ta) + X(ha): a E Q1 } of k-linear maps. A morphism f: X + Y between two representations of Q is a family {f(i): X(i)+ Y(i): in QO} of k-linear maps satisfying f(ha)oX(a) = Y(a)of(ta) for all arrows a. We denote the category of representations of Q by mod Q. It is an abelian category of global dimension at most 1. We will denote the only possibly non-trivial extension group Ext’ by Ext. The dimension vector of a representation X of Q is the vector dim X= (dim X(l), . . . . dim X(n)) in N”, and the dimension of X is the natural number dim X= dim X( 1) + . . . + dim X(n). The support of X is the full subquiver of Q whose vertices are {i: X(i) ZO}. For a vector d = (d,, . . . . d,) E N”, we define R(Q, d) to be the set of representations X of Q such that X(i) = k” for ie QO. Thus R(Q, d) is the product R(Q, d) = n Hom,(kdru, kdhe).