Tree Normal Forms for Quiver Representations

  • Ryan Kinser

    Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, USA
  • Thorsten Weist

    Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42097 Wuppertal, Germany
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We explore methods for constructing normal forms of indecomposable quiver representations. The first part of the paper develops homological tools for recursively constructing families of indecomposable representations from indecomposables of smaller dimension vector. This is then specialized to the situation of tree modules, where the existence of a special basis simplifies computations and gives nicer normal forms. Motivated by a conjecture of Kac, we use this to construct cells of indecomposable representations as deformations of tree modules. The second part of the paper develops geometric tools for constructing cells of indecomposable representations from torus actions on moduli spaces of representations. As an application, we combine these methods to construct families of indecomposables grouped into cells. These actually give a normal form for all indecomposables of certain roots.

Cite this article

Ryan Kinser, Thorsten Weist, Tree Normal Forms for Quiver Representations. Doc. Math. 24 (2019), pp. 1245–1294

DOI 10.4171/DM/704