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The (cyclic) enhanced nilpotent cone via quiver representations

  • Gwyn Bellamy
  • Magdalena BoosEmail author
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Abstract

The \({{\,\mathrm{GL}\,}}(V)\)-orbits in the enhanced nilpotent cone \(V\times \mathcal {N}\) are (essentially) in bijection with the orbits of a certain parabolic \(P\subseteq {{\,\mathrm{GL}\,}}(V)\) (the mirabolic subgroup) in the nilpotent cone \(\mathcal {N}\). We give a new parameterization of the orbits in the enhanced nilpotent cone, in terms of representations of the underlying quiver. This parameterization generalizes naturally to the enhanced cyclic nilpotent cone. Our parameterizations are different from the previous ones that have appeared in the literature. Explicit translations between the different parametrizations are given.

Mathematics Subject Classification

Primary 16G20 Secondary 16G60 17B08 

Notes

Acknowledgements

The authors would like to thank C. Johnson for his very precise and valuable ideas regarding the translation between our parametrization and his original parametrization of orbits. We also thank K. Bongartz and M. Reineke for helpful remarks on the subject. The first author was partially supported by EPSRC Grant EP/N005058/1.

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© The Author(s) 2018
corrected publication 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of GlasgowGlasgowUK
  2. 2.Fakultät für MathematikRuhr-Universität Bochum, Universitätsstraße 150BochumGermany

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