Motivic (Representation) Stability of Representation Varieties and Character Stacks
Márton Hablicsek, Jesse Vogel
TL;DR
This work develops a motivic analogue of representation stability by placing representation varieties and character stacks into Grothendieck rings and Ekedahl's stack variant, then analyzes their asymptotics under genus, free-group rank, and rank growth. The authors formulate conjectures for surface groups, free groups, and free abelian groups, and verify them in several cases (notably $G=\mathrm{GL}_r$, $\mathrm{SL}_r$, and certain upper-triangular groups) using motivic decompositions, induction/restriction, and topological quantum field theory techniques. Key contributions include a detailed framework for motivic stability across varieties and stacks, a decomposition by representations, and explicit computations for commuting tuples that illuminate when motivic representation stability holds or fails. The results advance the understanding of how algebraic and cohomological representations stabilize motivically and connect to classical stability theorems, offering algebraic tools for studying representation varieties and character stacks in high-rank or high-genus regimes.
Abstract
In this paper, we introduce the notions of motivic representation stability that is an algebraic counterpart of the notion of representation stability. In the process, we also introduce the notion of motivic decomposition for varieties equipped with an action of a finite group $G$. This motivic decomposition decomposes the virtual class of the variety with respect to irreducible rational representations of $G$. We also formulate conjectures on motivic representation stability in the context of representation varieties and character stacks, and we verify the conjectures for groups whose virtual classes have been extensively studied.