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On the motive of quotients of induced actions

Lucas de Amorin

TL;DR

The paper develops and applies a robust equivariant motivic framework for cyclic group actions to compute motives of quotients in the Grothendieck ring, with a focus on isotypic decompositions and permutation actions. It extends Vogel’s motivic construction to the setting of torus-knot representation varieties, enabling explicit calculations for GL$_r$ and SL$_r$ irreducible representation spaces, including nontrivial localization by $q$ and $q^i-1$. A core fibration method is refined to be equivariantly locally trivial, which, together with Möbius inversion and localization techniques, yields explicit formulas and correction terms for low ranks ($r o 1,2,3,4$). The results demonstrate precise GL/SL relationships in coprime cases and delineate the necessary corrections when coprimality fails, providing concrete motivic data in settings of interest to knot theory and algebraic group representations. These methods offer a pathway to compute motives of a broad class of character/representation varieties beyond the previously available $E$-polynomial or cohomological data, with potential applications to topology and geometric representation theory.

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.González-Prieto and V.Muñoz.

On the motive of quotients of induced actions

TL;DR

The paper develops and applies a robust equivariant motivic framework for cyclic group actions to compute motives of quotients in the Grothendieck ring, with a focus on isotypic decompositions and permutation actions. It extends Vogel’s motivic construction to the setting of torus-knot representation varieties, enabling explicit calculations for GL and SL irreducible representation spaces, including nontrivial localization by and . A core fibration method is refined to be equivariantly locally trivial, which, together with Möbius inversion and localization techniques, yields explicit formulas and correction terms for low ranks (). The results demonstrate precise GL/SL relationships in coprime cases and delineate the necessary corrections when coprimality fails, providing concrete motivic data in settings of interest to knot theory and algebraic group representations. These methods offer a pathway to compute motives of a broad class of character/representation varieties beyond the previously available -polynomial or cohomological data, with potential applications to topology and geometric representation theory.

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.González-Prieto and V.Muñoz.

Paper Structure

This paper contains 21 sections, 58 theorems, 240 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rational representations of cyclic groups
  4. Grothendieck ring of varieties
  5. Möbius inversion
  6. Motivic isotypic decompositions
  7. Definition and basic properties
  8. Example: fixed rank matrices
  9. Permutation of factors action
  10. Example: conjugacy classes in the general linear group
  11. Localization
  12. Character varieties of torus knots
  13. The core fibration
  14. Character vs representation varieties
  15. Rank two
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any two cyclic groups $\Gamma\subset\Gamma'$ there exists a unique function $\operatorname{Per}_\Gamma^{\Gamma'}:K_0^\Gamma(\mathrm{Var}_\mathbb{C})\to K_0^{\Gamma'}(\mathrm{Var}_\mathbb{C})$ such that:

Theorems & Definitions (113)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.1
  • ...and 103 more