Commuting Schemes of Upper Triangular Matrices and Their Connection to Representation Homology: Exploring Complete Intersection Properties

Core Concepts

This paper investigates the relationship between the representation homology of a Riemann surface with coefficients in a group of upper triangular matrices and the complete intersection properties of the corresponding commuting schemes.

Abstract

Bibliographic Information

Li, G. (2024). Commuting Schemes of Upper Triangular Matrices and Representation Homology [Preprint]. arXiv:2403.13953v2.

Research Objective

This paper aims to determine when commuting schemes of upper triangular matrices form a complete intersection by examining their connection to the representation homology of Riemann surfaces.

Methodology

The author utilizes tools from derived algebraic geometry, specifically representation homology, to analyze the algebraic structure of commuting schemes. They employ Koszul complexes and regular sequences to characterize complete intersections. Additionally, the author uses the Macaulay2 software package RepHomology for computational verification of their theoretical findings.

Key Findings

• The paper establishes an equivalence between the vanishing of higher-degree representation homology of a Riemann surface with coefficients in a unipotent upper triangular matrix group and the complete intersection property of the corresponding commuting scheme.
• Through computational analysis using Macaulay2, the author demonstrates that the commuting schemes C(U2), C(U3), C(U4), C(U5), C(B2), and C(B3) are complete intersections.
• Conversely, the paper proves that C(Un) is not a complete intersection for n ≥ 6, providing a counterexample to the general expectation.

Main Conclusions

The research demonstrates a novel connection between representation homology and the geometric properties of commuting schemes. The vanishing of higher representation homology provides a sufficient and necessary condition for a commuting scheme of upper triangular matrices to be a complete intersection.

Significance

This work contributes to the understanding of both representation homology and commuting schemes, highlighting the interplay between derived algebraic geometry and classical algebraic geometry. The findings have implications for the study of representation spaces and their geometric properties.

Limitations and Future Research

The paper focuses specifically on upper triangular matrices. Further research could explore similar connections between representation homology and commuting schemes for other types of matrix groups or more general algebraic groups. Investigating the properties of representation homology for non-complete intersection commuting schemes could also yield valuable insights.

Stats

Condition 1.1 holds for n ≤5, so that C(U2), C(U3), C(U4), C(U5) are complete intersections of correct dimensions.
In contrast, C(Un) is not a complete intersection of codimension (n−2)(n−1)/2 in Un × Un when n ≥6.

Quotes

Key Insights Distilled From

Commuting Schemes of Upper Triangular Matrices and Representation Homology

by Guanyu Li at arxiv.org 11-12-2024

https://arxiv.org/pdf/2403.13953.pdf

Deeper Inquiries

Can the methods used in this paper be extended to study commuting schemes of other types of algebraic groups beyond upper triangular matrices?

Extending the methods used in the paper to study commuting schemes of more general algebraic groups presents several challenges and potential avenues for exploration:
Challenges:

Regularity of the conunit kernel: The proofs of Proposition 3.1 and 3.2 heavily rely on the fact that the kernel of the counit map for Un and Bn is generated by a regular sequence (Lemma 2.5). This property might not hold for more general algebraic groups, making it difficult to directly apply the Koszul complex approach for computing representation homology.
Complexity of relations: For groups beyond upper triangular matrices, the relations defining the commuting scheme (i.e., the equations ensuring matrices commute) can become significantly more intricate. This complexity can make it harder to analyze the geometry of the scheme and determine properties like complete intersection.
Computational limitations: Even for relatively simple groups like U6, determining whether the commuting scheme is a complete intersection computationally is non-trivial. As the complexity of the group increases, the computational challenges in calculating representation homology and analyzing the geometry of the commuting scheme grow considerably.
Potential Avenues for Extension:

Subgroups and quotients: One approach could be to focus on subgroups or quotients of more general groups that might still admit a tractable description of their conunit kernel or have simpler relations defining their commuting schemes.
Alternative resolutions: Instead of relying solely on Koszul complexes, exploring alternative resolutions for computing representation homology might be fruitful. This could involve techniques from derived algebraic geometry or other homological methods.
Geometric insights: Even if complete intersection is not achievable, studying the structure of representation homology for more general commuting schemes could still provide valuable geometric insights. Analyzing the non-vanishing homology groups and their relations to the defining equations of the scheme could reveal information about its singularities, irreducible components, or other geometric properties.

What can be said about the structure of representation homology for commuting schemes that are not complete intersections, and does it provide any insights into their geometric properties?

When a commuting scheme is not a complete intersection, its representation homology becomes more intricate, and the vanishing condition in higher degrees no longer holds. Analyzing this more complex homology structure can still offer valuable insights into the geometry of the scheme:

Non-vanishing homology: The presence of non-vanishing homology groups in degrees higher than expected (as in Proposition 3.1 and 3.2) indicates the presence of "higher-order" relations among the defining equations of the commuting scheme. These relations reflect the failure of the scheme to be a complete intersection and can provide clues about its singularities.
Syzygies and free resolutions: The non-vanishing homology groups can be interpreted as syzygies in a free resolution of the coordinate ring of the commuting scheme. Studying these syzygies can shed light on the structure of the ideal defining the scheme and its singularities.
Deviations from complete intersection: The "size" and "pattern" of the non-vanishing homology groups can quantify how far the commuting scheme deviates from being a complete intersection. This information can be used to compare the complexity of different commuting schemes and understand how their geometric properties differ.
Example:
In the case of C(U6), which is not a complete intersection, analyzing the non-vanishing representation homology groups could reveal:

The minimal number of generators needed to describe the ideal defining C(U6).
The degrees in which these generators appear, providing information about the "order" of relations among the commuting equations.
Potential connections between the non-vanishing homology groups and specific geometric features of C(U6), such as the presence of particular singular loci.

How does the connection between representation homology and commuting schemes relate to other areas of mathematics, such as geometric representation theory or the study of moduli spaces?

The connection between representation homology and commuting schemes has rich implications for various areas of mathematics, particularly geometric representation theory and the study of moduli spaces:
Geometric Representation Theory:

Character varieties: Commuting schemes are closely related to character varieties, which parametrize representations of a group into a fixed algebraic group. Representation homology provides a powerful tool for studying the topology and geometry of character varieties, especially their cohomology rings and their relationship to the underlying group and its representations.
Moduli of local systems: Representation homology can be interpreted as the cohomology of moduli spaces of local systems on a topological space. These moduli spaces play a crucial role in gauge theory, and understanding their topology through representation homology has implications for studying gauge-theoretic invariants.
(Geometric) Langlands program: The study of commuting schemes and their representation homology has connections to the (geometric) Langlands program, which seeks to relate representations of a group to objects in algebraic geometry. Representation homology provides a bridge between these two worlds, offering insights into the geometric structures underlying representation theory.
Moduli Spaces:

Deformation theory: Commuting schemes can often be viewed as special cases of more general moduli spaces, such as moduli spaces of Higgs bundles or moduli spaces of flat connections. Representation homology can be used to study the deformation theory of these moduli spaces, providing information about their local structure and how they vary under perturbations.
Enumerative geometry: In some cases, representation homology can be used to compute enumerative invariants associated with moduli spaces, such as Donaldson-Thomas invariants or Gromov-Witten invariants. These invariants provide a quantitative measure of certain geometric objects within the moduli space.
Homotopical methods: The use of representation homology introduces powerful homotopical methods into the study of moduli spaces. This perspective can lead to new insights and connections with other areas of mathematics, such as homotopy theory and derived algebraic geometry.