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Commuting Schemes of Upper Triangular Matrices and Representation Homology

Guanyu Li

TL;DR

The paper connects generalized commuting schemes $C_g(U_n)$ to the representation homology $HR_*(\Sigma_g,U_n)$ of a genus-$g$ surface, proving that $C_g(U_n)$ is a global complete intersection if and only if $HR_i(\Sigma_g,U_n)=0$ for all $i\ge n$, with analogous results for the Borel case $C_g(B_n)$. Using a Koszul-complex framework, the authors derive explicit criteria and show that $HR_*(\Sigma_g,U_n)$ vanishes in high degrees for $n\le5$, yielding complete-intersection results for $C(U_n)$ with $n=2,3,4,5$, and similarly for $C(B_n)$ with $n=2,3$. They further provide a counterexample $C(U_6)$ that is not a complete intersection, highlighting a sharp boundary in the phenomenon. The work combines derived-algebraic techniques with explicit computations (via Macaulay2) to illuminate when classical geometric properties of commuting schemes align with vanishing-derived invariants, and it demonstrates the special nature of unipotent and Borel coefficients in this context.

Abstract

We establish a connection between generalised commuting schemes $C_g(U_n)$ of higher genus $g$, which are associated with a group scheme $U_n$ consisting of upper triangular unipotent matrices, and the representation homology $HR_*(Σ_g,U_n)$ of a Riemann surface $Σ_g$ with coefficients in the group $U_n$. As an outcome, we provide a numerical criterion for determining whether commuting schemes $C(U_n)$ form a complete intersection.

Commuting Schemes of Upper Triangular Matrices and Representation Homology

TL;DR

The paper connects generalized commuting schemes to the representation homology of a genus- surface, proving that is a global complete intersection if and only if for all , with analogous results for the Borel case . Using a Koszul-complex framework, the authors derive explicit criteria and show that vanishes in high degrees for , yielding complete-intersection results for with , and similarly for with . They further provide a counterexample that is not a complete intersection, highlighting a sharp boundary in the phenomenon. The work combines derived-algebraic techniques with explicit computations (via Macaulay2) to illuminate when classical geometric properties of commuting schemes align with vanishing-derived invariants, and it demonstrates the special nature of unipotent and Borel coefficients in this context.

Abstract

We establish a connection between generalised commuting schemes of higher genus , which are associated with a group scheme consisting of upper triangular unipotent matrices, and the representation homology of a Riemann surface with coefficients in the group . As an outcome, we provide a numerical criterion for determining whether commuting schemes form a complete intersection.
Paper Structure (13 sections, 15 theorems, 80 equations)

This paper contains 13 sections, 15 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Notation and conventions
  4. Acknowledgment
  5. Preliminaries
  6. Koszul complex and codimension
  7. Group schemes and the grading
  8. Commuting schemes and higher genus generalisations
  9. Representation schemes
  10. Representation homology
  11. Main Results
  12. Proofs of the main results
  13. A counterexample

Key Result

Theorem 1.1

Let $U_n$ be the algebraic group of $n\times n$ upper triangular unipotent matrices over a characteristic $0$ field $k$, and let $C_g(U_n)$ be its commuting scheme of genus $g$ (see Def_CommSchHigherGenus). Then $C_g(U_n)$ is a (global) complete intersection if and only if the representation homolog where $\Sigma_g$ is the Riemann surface of genus $g$. In the $g=1$ case, condition Eq_Vanishing hol

Theorems & Definitions (44)

  • Theorem 1.1: \ref{['Proposition_UnipotnetGp']}, \ref{['Thm_List']} and \ref{['Proposition_U6NonCI']}
  • Definition 2.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Theorem 2.2
  • Corollary 2.2.1
  • Definition 2.3
  • Corollary 2.2.2
  • Definition 2.4
  • ...and 34 more