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Presentations of representations

Daniel Minahan, Andrew Putman

TL;DR

The paper introduces a flexible, representation-theoretic framework for constructing presentations by generators and relations for finite-dimensional representations of $\mathrm{SL}_n(\mathbb{Z})$ and $\mathrm{Sp}_{2g}(\mathbb{Z})$, with a focus on infinite generator-relations that become finite through a linearization map. It establishes precise isomorphism results for a range of representations, including the standard and adjoint representations of $\mathrm{SL}_n(\mathbb{Z})$, the standard and kernel representations of $\mathrm{Sp}_{2g}(\mathbb{Z})$, and several symmetric/antisymmetric variants (e.g., symmetric kernel and isotropic data). A core methodological theme is a three-step outline: (1) pick a natural generating set mapping to a basis; (2) show the orbit of generators spans the target representation; (3) verify group actions preserve the span, enabling an isomorphism via the linearization map. The work further develops a substantial refinement program for the symmetric kernel by leveraging isotropic and strong isotropic pairs, culminating in isomorphisms to contraction-induced kernels such as $\mathcal{K}_g^a$ and $\mathcal{K}_g^s$; these technical advances underpin applications to the Torelli group's second homology and related algebraic structures. The results provide a coherent, extensible toolkit to translate between algebraic presentations and representation-theoretic data, with implications for understanding homological properties of mapping class groups and related automorphism groups of lattices. The authors also discuss broad generalizations and the potential for a unifying abstract theorem that specializes to all established and future variants.

Abstract

We give a new technique for constructing presentations by generators and relations for representations of groups like $SL_n(\mathbb{Z})$ and $Sp_{2g}(\mathbb{Z})$. Our results play an important role in recent work of the authors calculating the second homology group of the Torelli group.

Presentations of representations

TL;DR

The paper introduces a flexible, representation-theoretic framework for constructing presentations by generators and relations for finite-dimensional representations of and , with a focus on infinite generator-relations that become finite through a linearization map. It establishes precise isomorphism results for a range of representations, including the standard and adjoint representations of , the standard and kernel representations of , and several symmetric/antisymmetric variants (e.g., symmetric kernel and isotropic data). A core methodological theme is a three-step outline: (1) pick a natural generating set mapping to a basis; (2) show the orbit of generators spans the target representation; (3) verify group actions preserve the span, enabling an isomorphism via the linearization map. The work further develops a substantial refinement program for the symmetric kernel by leveraging isotropic and strong isotropic pairs, culminating in isomorphisms to contraction-induced kernels such as and ; these technical advances underpin applications to the Torelli group's second homology and related algebraic structures. The results provide a coherent, extensible toolkit to translate between algebraic presentations and representation-theoretic data, with implications for understanding homological properties of mapping class groups and related automorphism groups of lattices. The authors also discuss broad generalizations and the potential for a unifying abstract theorem that specializes to all established and future variants.

Abstract

We give a new technique for constructing presentations by generators and relations for representations of groups like and . Our results play an important role in recent work of the authors calculating the second homology group of the Torelli group.

Paper Structure

This paper contains 164 sections, 106 theorems, 570 equations.

Table of Contents

  1. Introduction
  2. Special linear group, standard representation
  3. Adjoint representation
  4. Symplectic group, standard representation
  5. Symplectic kernel
  6. Symmetric square
  7. Quotient representation
  8. Symmetric contraction
  9. Symmetric kernel
  10. Symplectic pairs
  11. Symmetric kernel presentation
  12. Symmetric square, II
  13. Final remarks
  14. Notation and conventions
  15. Outline
  16. ...and 149 more sections

Key Result

Theorem A

For $n \geq 2$, the linearization map $\Phi\colon \mathfrak{Q}_n \rightarrow \mathbb{Q}^n$ is an isomorphism.

Theorems & Definitions (290)

  • Definition 1.1
  • Theorem A
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • Theorem B
  • Remark 1.7
  • Definition 1.8
  • ...and 280 more