Geometric Invariant Theory and Moduli Spaces of Quiver Representations

Date

2025-8-28

Author

Türker, Anıl Berkcan

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The study of quiver representations is deeply motivated by several fundamental results: The Happel--Reiten--Smalø theorem shows that the following are hereditary abelian categories: 1) quiver representations, 2) coherent sheaves on smooth curves, and 3) certain finite exceptions such as coherent sheaves on wild hereditary curves. Furthermore, the Gabriel--Mitchell theorem establishes that any finite-dimensional $k$-algebra is Morita equivalent to a path algebra with admissible relations. Consequently, understanding quiver representations provides significant insight into the objects of various categories. Classification problems in representation theory naturally lead to constructing moduli spaces, where isomorphism classes correspond to orbits under appropriate group actions. For quiver representations, the isomorphism classes coincide with orbits of the conjugation action of some general linear group on some affine space. This connection allows us to employ geometric invariant theory (GIT) effectively—particularly powerful when considering reductive groups like general linear groups and proper linearizations with ample bundles. This thesis surveys the rich interplay between quiver representations and GIT, provides necessary technical details, and suggests potential future directions in the field.

Subject Keywords

quiver, quiver representation, quiver variety, moduli space, algebraic group, geometric invariant theory, linearisation, vector bundles

URI

https://hdl.handle.net/11511/116632

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Graduate School of Natural and Applied Sciences, Thesis