On the motive of quotients of induced actions

TL;DR

This paper develops computational methods to determine the motives of cyclic quotients and applies these tools to compute the motives of certain representation varieties related to torus knots and linear groups.

Contribution

It introduces new computational techniques for motives of cyclic quotients and extends existing strategies to broader classes of representation varieties.

Findings

Computed motives of low rank representation varieties for torus knots.

Established an equivariant approach for motives of linear group representations.

Provided explicit examples illustrating the effectiveness of the methods.

Abstract

We explore computational tools that allow to compute the class on the Grothendieck ring of varieties of finite cyclic quotients in some interesting examples. As an main application, we determine the motive of low rank representation varieties associated with torus knots and general linear groups using an equivariant analogue of the strategy for special linear groups due to A.Gonz\'alez-Prieto and V.Mu\~noz.