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 Title: Stratifications of representations and cyclic quivers Author(s): Ochoa de Alaiza Gracia, Itziar Director of Research: Nevins, Thomas Doctoral Committee Chair(s): Loja Fernandes, Rui Doctoral Committee Member(s): Yong, Alexander; Dodd, Christopher Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(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 $Q$. Issue Date: 2018-07-06 Type: Text URI: http://hdl.handle.net/2142/101520 Rights Information: Copyright 2018 by Itziar Ochoa de Alaiza Gracia. All rights reserved. Date Available in IDEALS: 2018-09-27 Date Deposited: 2018-08