We define a notion of representation stability for consistent sequences of $\glnfq$-modules. We prove that the consistent sequence obtained from a finitely generated VI-module or a finitely generated VIC-module is representation stable.

We define a functorial way to turn a cyclic operad into a dioperad. Bergman's Diamond Lemma and Hoffbeck's PBW bases for operads are generalized to the setting of dioperads. We use these results to show that the induced dioperads from the cyclic operads

Com and Lie are Koszul.

The category $\FI_G$ has been studied in relation to the representation theory of the wreath products $G \wr S_n$, where $G$ is a finite group. We take $G=A$ to be abelian and define the category $\FI_A^H$ associated to certain subgroups $J_n$ of $A \wr S_n$, including the finite complex reflection groups $G(m,p,n)$ defined by Shepard and Todd. We also give axioms for ``$\FI$-like'' categories $\C$, and prove the equivalence of the noetherian property for $\C$-modules over $k$ with the noetherian property for $\A$-modules over $k$, where $\A$ is a suitably restricted subcategory of $\C$ and $k$ is a commutative noetherian ring with unity. We apply this result to show that representation stability for a sequence of finite-dimensional $J_n$-representations over $\mathbb{C}$ is equivalent to finite generation of the corresponding $\FI_A^H$-module. We also prove the analogous result for representations of the alternating groups. Lastly, we prove homological stability for the family $\{J_n\}$ with twisted coefficients from a finitely generated $\FI_A^H$-module over $\Z$, as well as the equivalence of certain Serre quotient categories of locally noetherian $\C$- and $\A$-modules.

Twisted graded Hecke algebras were introduced by S. Witherspoon as a common generalization of graded Hecke algebras and twisted symplectic reflection algebras. In this thesis, the structure and representation theory of twisted graded Hecke algebras for elementary abelian groups are studied. Such an algebra A is finitely generated as a module over its center. Moreover, for a generic central character χ, there exists a unique simple A-module on which the center acts by χ.

We introduce a category FI$_{G, A}$ which depends upon a group $G$ and a representation $A$ of $G$. We establish Noetherian and homological properties of FI$_{G, A}$-modules. We also study the representation stability of FIM-modules.

A development of Auslander-Reiten theory in the language of stable infinity-categories is presented. An infinity-category is a special kind of simplicial set which provides a common generalization of ordinary categories and nice topological spaces. Higher Auslander-Reiten theory can therefore be understood as a homotopy-theoretic analogue of the classical theory, which has proved to be an indispensable tool in many areas of representation theory.

We begin by introducing almost-split and irreducible morphisms in infinity-categories and establish their basic properties in direct analogy with the classical notions. We go on to describe morphisms determined by objects, a generalization of almost-split morphisms originally formulated by Auslander that until recently received little attention. We prove, using Brown representability, that every collection of maps with domain a compact object C uniquely determines, up to homotopy, a minimal C-determined morphism. From this result, the existence of almost-split and irreducible morphisms can be deduced.

We next describe the analogues of almost-split sequences in stable infinity-categories and prove that they exist in any compactly generated stable infinity-category with sufficiently small objects. Building on our earlier study of morphisms determined by objects, we then introduce Auslander functors on stable infinity-categories and show that they always exist on compactly generated stable infinity-categories. In good circumstances, Auslander functors specialize to Serre functors. This observation leads to a general duality formula which specializes to the classical Auslander-Reiten formula on the homotopy category.

Finally, we focus on an important class of examples of compactly generated stable infinity-categories associated to any Noetherian algebra over a complete local Noetherian ring. In this situation, we give a construction of an Auslander-Reiten translation functor and explain how it recovers the classical Auslander-Reiten translation on the associated (triangulated) homotopy category.