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Finite Parabolic Conjugation on Varieties of Nilpotent Matrices

Abstract

We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of GLn(C) on the variety of x-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of 2-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. We show that in all non-finite cases, the corresponding quiver algebra is of wild representation type.

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Correspondence to Magdalena Boos.

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Presented by Peter Littelmann.

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Boos, M. Finite Parabolic Conjugation on Varieties of Nilpotent Matrices. Algebr Represent Theor 17, 1657–1682 (2014). https://doi.org/10.1007/s10468-014-9464-0

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Keywords

  • Parabolic orbits of nilpotent matrices
  • Degenerations
  • Finite classification

Mathematics Subject Classification (2010)

  • 16G20
  • 16G60
  • 16W22