Stratifications of representations and cyclic quivers

Ochoa de Alaiza Gracia, Itziar

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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

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

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

Copyright and License Information

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

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