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28 March 2024
 
  » arxiv » 1509.8051

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Irreducible components of varieties of representations II
Birge Huisgen-Zimmermann ; Ian Shipman ;
Date 27 Sep 2015
AbstractThis article is part of a program to evolve the generic representation theory of basic finite dimensional algebras A over an algebraically closed field K, in other words, the goal is to determine the irreducible components of the varieties Rep(A,d) parametrizing the finite dimensional representations with dimension vector d, and to generically describe the representations encoded by the components. Here we primarily target truncated path algebras, i.e., algebras of the form A = KQ/I for a quiver Q, where I is generated by all paths of some fixed length in the path algebra KQ. The main result characterizes the irreducible components of the affine (resp. projective) parametrizing variety Rep(A,d) (resp. GRASS_d(A)) in case Q is acyclic. Our classification is in representation-theoretic terms, permitting to list the components from the quiver and Loewy length of A. Combined with existing theory, it moreover yields an array of generic features of the modules parametrized by the irreducible components, such as generic minimal projective presentations, generic skeleta ("path bases" recruited from a finite set of eligible paths), generic dimensions of endomorphism rings, generic socles, etc.
The information on truncated path algebras with acyclic quiver supplements the comparatively well-developed theory available in the special case where A is hereditary, i.e., for I = 0: On one hand, we add to the classical generic results regarding the d-dimensional KQ-modules, they address only the modules of maximal Loewy length. On the other hand, the more general theory for I nonzero developed here fills in generic data on the d-dimensional KQ-modules of any fixed Loewy length.
Source arXiv, 1509.8051
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