Irreducible components of varieties of representations II

Abstract

The goals of this article are as follows: (1) To determine the irreducible components of the affine varieties parametrizing the representations of $ \Lambda $ with dimension vector d, where $ \Lambda $ traces a major class of finite dimensional algebras; (2) To generically describe the representations encoded by the components. The target class consists of those truncated path algebras $ \Lambda $ over an algebraically closed field K which are based on a quiver Q without oriented cycles. The main result characterizes the irreducible components of the representation variety in representation-theoretic terms and provides a means of listing them from quiver and Loewy length of $ \Lambda $. Combined with existing theory, this classification moreover yields an array of generic features of the modules parametrized by the components, such as generic minimal projective presentations, generic sub- and quotient modules, etc. Our second principal result pins down the generic socle series of the modules in the components; it does so for more general $ \Lambda $, in fact. The information on truncated path algebras of acyclic quivers supplements the theory available in the special case where $ \Lambda = KQ $, filling in generic data on the d-dimensional representations of Q with any fixed Loewy length.