${\mathrm{FI}}_G$-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 $(G^n⋊S_n)$-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 ${\tilde{A}}_n$ and ${\tilde{C}}_n$ -the cohomology of Fouxe-Rabinowitsch groups and many more examples.

33 pages; this is the first of two papers on FI_G modules; the forthcoming second paper is focused on arithmetic statistics

References in corpus (13)

• Representation theory and homological stability
• FI-modules over Noetherian rings
• Stabilization for mapping class groups of 3-manifolds
• On the quotient ring by diagonal harmonics
• Homology of FI-modules
• Representations of categories of G-maps
• Homological Invariants of FI-modules and FI_G-modules
• Periodicity in the cohomology of symmetric groups via divided powers
• On the degree-wise coherence of FI_G-modules
• FI_W-modules and stability criteria for representations of the classical Weyl groups
• Homology of braid groups, the Burau representation, and ${\mathbb{F}}_q$-points on superelliptic curves
• Coinduction functor in representation stability theory
• Two homological proofs of the Noetherianity of $F{I}_G$

Cited by in corpus (5)

• Representations of categories of G-maps
• Categories of FI type: a unified approach to generalizing representation stability and character polynomials
• A generating function approach to new representation stability phenomena in orbit configuration spaces
• ${\mathrm{FI}}_G$-modules and arithmetic statistics
• Asymptotic behaviors of representations of graded categories with inductive functors