Brauner, Sarah2023-09-192023-09-192023-05https://hdl.handle.net/11299/257026University of Minnesota Ph.D. dissertation .May 2023. Major: Mathematics. Advisor: Victor Reiner. 1 computer file (PDF); viii, 211 pages.The overarching theme of this thesis is to study monoid algebras and cohomology rings of configuration spaces through the lens of symmetry, with the goal of answering questions, forging connections and translating methods between algebraic combinatorics, representation theory and algebraic topology. Chapters 3 and 4 of the thesis establish novel connections between cohomology rings of configuration spaces and classical combinatorial algebras arising in the theory of reflection groups and hyperplane arrangements. These chapters introduce new tools to describe the symmetries of several combinatorially significant topological spaces. Chapter 5 studies certain monoids called left regular bands from the perspective of invariant theory, developing a framework to study the free left regular band and its q-analogue in parallel.enalgebraic combinatoricsconfiguration spacesCoxeter groupsrepresentation theorySolomon's descent algebraThesis or Dissertation