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The equivariant cohomology ring of the representation variety $\Hom(\Z^2,\mathrm{GL}_n(\C))$

Simon Gritschacher

TL;DR

The paper provides a complete integral presentation of the GL_n(C)-equivariant cohomology ring of the commuting representation variety C_2(GL_n(C)) by realizing it as a saturated quotient of a free Λ_n-algebra, with Λ_n the invariant ring of the maximal torus, and generators coming from Chern-theoretic data. It proves torsion-freeness, identifies a minimal generating set of size 3n, and derives the explicit Δ^2-saturated ideal of relations through determinant identities involving the Vandermonde matrix. It also yields a non-equivariant presentation over fields of characteristic zero or not dividing n!, and explains how inverting n! reduces complexity; the work further relates to gauge-theoretic maps in Atiyah–Bott theory and establishes integral surjectivity of the associated map in genus 1. The methods blend Hochschild/Loday theory, invariant theory, and precise algebraic manipulations to obtain a fully explicit, structured description of the equivariant cohomology ring and its reductions.

Abstract

We give a presentation of the $\mathrm{GL}_n(\C)$-equivariant cohomology ring with $\Z$-coefficients of the variety $\Hom(\Z^2,\mathrm{GL}_n(\C))\subseteq \mathrm{GL}_n(\C)^2$ for any $n$. It is torsion free and minimally generated as a $H^\ast B\mathrm{GL}_n(\C)$-algebra by $3n$ elements. The ideal of relations is the saturation of an $n$-generator ideal by even powers of the Vandermonde polynomial. For coefficients in a field whose characteristic does not divide $n!$, we also give a presentation of the non-equivariant cohomology ring of $\Hom(\Z^2,\mathrm{GL}_n(\C))$.

The equivariant cohomology ring of the representation variety $\Hom(\Z^2,\mathrm{GL}_n(\C))$

TL;DR

The paper provides a complete integral presentation of the GL_n(C)-equivariant cohomology ring of the commuting representation variety C_2(GL_n(C)) by realizing it as a saturated quotient of a free Λ_n-algebra, with Λ_n the invariant ring of the maximal torus, and generators coming from Chern-theoretic data. It proves torsion-freeness, identifies a minimal generating set of size 3n, and derives the explicit Δ^2-saturated ideal of relations through determinant identities involving the Vandermonde matrix. It also yields a non-equivariant presentation over fields of characteristic zero or not dividing n!, and explains how inverting n! reduces complexity; the work further relates to gauge-theoretic maps in Atiyah–Bott theory and establishes integral surjectivity of the associated map in genus 1. The methods blend Hochschild/Loday theory, invariant theory, and precise algebraic manipulations to obtain a fully explicit, structured description of the equivariant cohomology ring and its reductions.

Abstract

We give a presentation of the -equivariant cohomology ring with -coefficients of the variety for any . It is torsion free and minimally generated as a -algebra by elements. The ideal of relations is the saturation of an -generator ideal by even powers of the Vandermonde polynomial. For coefficients in a field whose characteristic does not divide , we also give a presentation of the non-equivariant cohomology ring of .
Paper Structure (13 sections, 29 theorems, 155 equations)

This paper contains 13 sections, 29 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. Our main theorem
  3. Outline of the paper
  4. A spectral sequence and the absence of torsion
  5. Preliminaries
  6. Loday construction
  7. The spectral sequence
  8. Determination of a generating set
  9. Determination of the ideal of relations
  10. Minimality of the generating set
  11. Inverting $n!$
  12. Integral surjectivity of the Atiyah-Bott map
  13. $\Gamma$-spaces and the spectral sequence

Key Result

Theorem 1.1

For every $n\geq 0$ there is an isomorphism of $\Lambda_n$-algebras where $\mathcal{J}$ is the saturation of the ideal $(R_1,\dots,R_n)$ with respect to $\Delta^2$. Moreover, this algebra cannot be generated by fewer than $3n$ elements.

Theorems & Definitions (69)

  • Theorem 1.1
  • Example 2.2
  • Definition 2.3
  • Example 2.4
  • Example 2.5
  • Corollary 2.6
  • proof
  • Proposition 2.7
  • proof
  • Corollary 2.8
  • ...and 59 more