Motivic (Representation) Stability of Representation Varieties and Character Stacks

TL;DR

This paper introduces the concept of motivic representation stability for varieties with group actions, providing a new algebraic framework and conjectures for understanding the structure of representation varieties and character stacks.

Contribution

It develops the notion of motivic representation stability and motivic decomposition, extending the concept of stability to an algebraic and motivic setting, and formulates and verifies related conjectures.

Findings

Defined motivic representation stability and motivic decomposition.

Formulated conjectures on motivic stability for representation varieties.

Verified conjectures for specific groups with well-studied virtual classes.

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.