Optimal filtrations on representations of finite dimensional algebras

Abstract

We present a representation theoretic description of the non-empty strata in the Hesselink stratification of the nullcone of representations of quivers. We use this stratification to define optimal filtrations on representations of finite dimensional algebras. As an application we investigate the isomorphism problem for uniserial representations. 1. OPTIMAL FILTRATIONS 1.1. Let A be a finite dimensional algebra over an algebraically closed field k. In this paper we want to parameterize isomorphism classes of finite dimensional Amodules having a specific Jordan-Holder sequence. In particular, we want to relate the recent results of K. Bongartz and B. Huisgen-Zimmermann [3, 4, 1] on uniserial modules to the Hesselink stratification of nullcones. By Morita theory we may reduce to the case that A is a basic algebra. That is, all simple A-modules are one dimensional. In this case we can write A as the quotient of the path algebra of a quiver and relate finite dimensional A-modules to representations of this quiver. 1.2. A quiver Q is a directed graph on a finite set of vertices {vl,..., Vn}. Let aij be the number of directed arrows from vi to vj (or loops if v = vj). The Euler-form of Q is the bilinear form X: Z x Zn Z determined by the matrix X = (Xij)i,j c Mn(Z) with entries Xij = 6ij aij Clearly, X encodes the structure of the directed graph Q. A representation V of a quiver Q of dimension vector a = (al,..., an) E Nn assigns to every arrow vi -vj in Q a matrix V(Q) E Majixi (k). The set of all a-dimensional representations form an affine space,