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Author ORCID Identifier

https://orcid.org/0009-0003-1190-0012

Date Available

4-20-2026

Year of Publication

2026

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

Faculty

Khrystyna Serhiyenko

Faculty

Bertrand Guillou

Abstract

In 2001, Fomin and Zelevinsky introduced cluster algebras which appear as coordinate rings of many varieties. We study cluster algebras coming from Richardson varieties. Leclerc gives a cluster structure on Richardson varieties using the representation theory of preprojective algebras. While this construction is very algebraic, we take a more combinatorial approach. The main goal is to find a combinatorial description for when certain cluster variables are compatible, or equivalently when modules defined by lattice paths in the preprojective algebra have trivial extensions. We extend the known results from Geiss, Leclerc, and Schröer that answer this question in the case of the Grassmannian by using various methods. We introduce the notion of extending a module, describe how add/remove operators applied to pairs of modules affects the extension space between them, and provide homological and combinatorial conditions that determine when two arbitrary modules have trivial extension.

Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2026.95

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