$\mathrm{FI}_G$-modules, orbit configuration spaces, and complex reflection groups

Abstract

The category $\mathrm{FI}_G$ was first defined and explored by Sam-Snowden. Here, we develop more of the machinery of $\mathrm{FI}_G$-modules and find numerous examples to apply it to, extending the work of Church-Ellenberg-Farb and Wilson. In particular we develop a notion of character polynomials for $\mathrm{FI}_G$-modules with $G$ finite, a notion of representation stability which we call $K_0$-stability even when $G$ is infinite virtually polycyclic, and apply the notion of finite presentation degree when $G$ is a general infinite group. We use this to analyze numerous families of $(G^n \rtimes S_n)$-modules, such as: -the cohomology and homotopy groups of orbit configuration spaces -the diagonal coinvariant algebra of complex reflection groups -the homology of affine pure braid groups of type $\widetilde{A}_n$ and $\widetilde{C}_n$ -the cohomology of Fouxe-Rabinowitsch groups and many more examples.