Stratifications of representations and cyclic quivers

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_{β\in\textsf{KN}} S_\beta detemined by 1-parameter subgroups β. 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 β'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 $Q$.

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

2018

Author dc:creator

• Ochoa de Alaiza Gracia, Itziar

Contributors dc:contributor

• Nevins, Thomas
• Loja Fernandes, Rui
• Yong, Alexander
• Dodd, Christopher

• Representations, quivers.

Statement dc:rights

• Copyright 2018 by Itziar Ochoa de Alaiza Gracia. All rights reserved.

Language dc:language

en

Handle dc:identifier

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

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F101520