University of Illinois Urbana-Champaign

Stratifications of representations and cyclic quivers

Ochoa de Alaiza Gracia, Itziar

Content Files
OCHOADEALAIZAGRACIA-DISSERTATION-2018.pdf

Permalink

https://hdl.handle.net/2142/101520

Description

Title
Stratifications of representations and cyclic quivers
Author(s)
Ochoa de Alaiza Gracia, Itziar
Issue Date
2018-07-06
Director of Research (if dissertation) or Advisor (if thesis)
Nevins, Thomas
Doctoral Committee Chair(s)
Loja Fernandes, Rui
Committee Member(s)
  • Yong, Alexander
  • Dodd, Christopher
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2018-09-27T16:17:36Z
Keyword(s)
Representations, quivers.
Abstract
Given an algebraic variety $X$ with an action of a reductive group $G$, geometric invariant theory splits $X$ as the disjoint union $X=X^{ss}\sqcup X^{un}$ of the semistable and unstable locus. The Kirwan-Ness stratification refines $X$ even more by describing $X^{un}$ as a disjoint union of strata $X^{un}=\displaystyle\sqcup_{\beta\in\textsf{KN}} S_\beta$ detemined by 1-parameter subgroups $\beta$. In this thesis we study the 1-parameter subgroups that determine the Kirwan-Ness stratifications of representations. We will describe an algorithm that finds the $\beta$'s and we show that such algorithm can be simplified when our space is of the form $T^*V$ where $V$ is a vector space. We go on to investigate more deeply the 1-parameter subgroups in the case of the space of representations $\text{Rep}(Q,v)$ of a cyclic quiver $Q$.
Graduation Semester
2018-08
Type of Resource
text
Permalink
http://hdl.handle.net/2142/101520
Copyright and License Information
Copyright 2018 by Itziar Ochoa de Alaiza Gracia. All rights reserved.

Owning Collections

Graduate Dissertations and Theses at Illinois PRIMARY
Graduate Theses and Dissertations at Illinois
Dissertations and Theses - Mathematics