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Understanding Finite Dimensional Representations Generically

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Geometric and Topological Aspects of the Representation Theory of Finite Groups (PSSW 2016)

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Abstract

We survey the development and status quo of a subject best described as “generic representation theory of finite dimensional algebras”, which started taking shape in the early 1980s. Let \({\Lambda }\) be a finite dimensional algebra over an algebraically closed field. Roughly, the theory aims at (a) pinning down the irreducible components of the standard parametrizing varieties for the \({\Lambda }\)-modules with a fixed dimension vector, and (b) assembling generic information on the modules in each individual component, that is, assembling data shared by all modules in a dense open subset of that component. We present an overview of results spanning the spectrum from hereditary algebras through the tame non-hereditary case to wild non-hereditary algebras.

Dedicated to Dave Benson on the occasion of his sixtieth birthday

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Authors and Affiliations

  1. Department of Mathematics, University of California, Santa Barbara, CA, 93106, USA

    K. R. Goodearl & B. Huisgen-Zimmermann

Authors
  1. K. R. Goodearl
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  2. B. Huisgen-Zimmermann
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Correspondence to B. Huisgen-Zimmermann .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, USA

    Jon F. Carlson

  2. Department of Mathematics, University of Utah, Salt Lake City, UT, USA

    Srikanth B. Iyengar

  3. Department of Mathematics, University of Washington, Seattle, WA, USA

    Julia Pevtsova

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Goodearl, K.R., Huisgen-Zimmermann, B. (2018). Understanding Finite Dimensional Representations Generically. In: Carlson, J., Iyengar, S., Pevtsova, J. (eds) Geometric and Topological Aspects of the Representation Theory of Finite Groups. PSSW 2016. Springer Proceedings in Mathematics & Statistics, vol 242. Springer, Cham. https://doi.org/10.1007/978-3-319-94033-5_6

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