University of Illinois Urbana-Champaign

On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution

Im, Mee Seong

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https://hdl.handle.net/2142/49392

Description

Title
On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution
Author(s)
Im, Mee Seong
Issue Date
2014-05-30T16:41:46Z
Director of Research (if dissertation) or Advisor (if thesis)
Nevins, Thomas A.
Doctoral Committee Chair(s)
Kedem, Rinat
Committee Member(s)
  • Nevins, Thomas A.
  • Bergvelt, Maarten J.
  • Schenck, Henry K.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • Algebraic geometry
  • representation theory
  • quiver varieties
  • filtered quiver variety
  • quiver flag variety
  • semi-invariant polynomials
  • invariant subring
  • Derksen-Weyman
  • Domokos-Zubkov
  • Schofield-van den Bergh
  • ADE-Dynkin quivers
  • affine Dynkin quivers
  • quivers with at most two pathways between any two vertices
  • filtration of vector spaces
  • classical invariant theory
  • the Hamiltonian reduction of the cotangent bundle of the enhanced Grothendieck-Springer resolution
  • almost-commuting varieties
  • affine quotient
Abstract
We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to conjugation by invertible (block) upper triangular matrices. With this notion in mind, we describe the ring of invariant polynomials for interesting families of quivers, namely, finite $ADE$-Dynkin quivers and affine type $\widetilde{A}$-Dynkin quivers. We then study their relation to an important and fundamental object in representation theory called the Grothendieck-Springer resolution, and we conclude by stating several conjectures, suggesting further research.
Graduation Semester
2014-05
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http://hdl.handle.net/2142/49392
Copyright and License Information
Copyright 2014 Mee Seong Im

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