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Abstract

This thesis sets out to explore and implement the paradigm of representation stability, specifically in the study of sequences of linear subspace arrangements, their stable combinatorics, topology and arithmetic. In a traditional sense, the sequences of arrangements in question do not exhibit any form of stability, e.g. the Betti numbers of their complements grow to infinity. But when one considers the symmetries at play and the various maps between the arrangements, a new notion of stability presents itself: representation stability, where a finite collection of patterns is merely translated around by increasingly large groups. In this sense, stability is understood as a notion of finite generation. One way to encode the structure of various symmetry groups and intertwining maps between them is using the language of diagrams. These are functors from a fixed category into the category of arrangements, vector spaces or any other target category. Thus a system of intertwined group actions can be treated as a single mathematical object, and finite generation gets a precise meaning. Representation stability therefore consists of two main aspects: identify finitely generated diagrams and operations that preserve finite generation (e.g. Noetherianity theorems), then extract stable invariants from a finitely generated diagram (e.g. polynomial characters). This work addresses both of the above aspects in a general axiomatic framework. We define and study finitely generated diagrams of linear subspace arrangements - these occur in many natural examples from algebraic geometry and combinatorics, such as colored configuration spaces, $k$-equals arrangements and covers of moduli spaces of rational maps. Such collections of arrangements exhibit finite generation in their intersection posets, and this in turn leads to finite generation in the cohomology of their complements. We then study the reprecautions of this finite generation using an adaptation of character theory to the study of diagrams. Lastly, we adapt the Grothendieck-Lefschetz fixed point formula to provide a bridge between cohomological representation stability results and asymptotic factorization statistics of orbits over finite fields.

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