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Mathematische Zeitschrift

, Volume 287, Issue 3–4, pp 1083–1107 | Cite as

Irreducible components of varieties of representations: the acyclic case

  • B. Huisgen-Zimmermann
  • I. Shipman
Article

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.

Keywords

Representations of finite dimensional algebras Quivers with relations Parametrizing varieties Irreducible components Generic properties of representations 

Mathematics Subject Classification

Primary 16G10 Secondary 16G20 14M99 

Notes

Acknowledgements

We wish to thank Eric Babson for numerous stimulating conversations on the subject of components at MSRI. Moreover, we thank the referee for his/her meticulous reading of the manuscript which led to significant improvements. The first author was partially supported by an NSF grant while carrying out this work. While in residence at MSRI, Berkeley, both authors were supported by NSF grant 0932078 000. The second author was also partially supported by NSF award DMS-1204733.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of MathematicsUniversity of UtahSalt Lake CityUSA

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