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

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Im, Mee Seong

Contributors dc:contributor

• Schenck, Henry K.
• Nevins, Thomas A.
• Kedem, Rinat
• Bergvelt, Maarten J.

• 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

Statement dc:rights

• Copyright 2014 Mee Seong Im

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/49392