Irreducible components of varieties of representations: the acyclic case

Abstract

The goals of this article are as follows: (1) To determine the irreducible components of the affine varieties ${\mathbf {Rep}}_{\mathbf {d}}(\Lambda )$ parametrizing the representations with dimension vector $\mathbf {d}$, 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 ${\mathbf {Rep}}_{\mathbf {d}}(\Lambda )$ 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 $\mathbf {d}$-dimensional representations of Q with any fixed Loewy length.