Dimension vectors of elementary modules of generalized Kronecker quivers

Let k be an algebraically closed field. The generalized or n-Kronecker quiver K(n) is the with two vertices, called a source and sink, n arrows from to sink. Given finite-dimensional module M of path algebra kK(n)=Kn, we consider its dimension vector dim¯M=(dimkM1,dimkM2). F={(x,y)|2nx≤y≤x}, let (x,y)∈F. We construct X(x, y) Kn, prove it elementary. Suppose that dim¯M=(x,y). show that:if elementary module, then x<2n, andwhen x+y=n+1, if only form y).