Deligne-Simpson Problem in Algebraic Geometry

• The Deligne-Simpson Problem is a central existence question that seeks irreducible representations with given conjugacy classes in GLₙ(ℂ), unifying algebraic geometry and differential equations.
• It employs quiver representation theory and root system criteria to translate monodromy conditions into geometric and combinatorial constraints essential for classifying moduli spaces.
• Its solutions facilitate the construction of rigid local systems and serve as a basis for analytic classifications of Fuchsian systems, impacting representation theory and nonabelian Hodge correspondence.

The Deligne–Simpson problem (DSP) is a central existence question in algebraic geometry, representation theory, and the theory of linear differential equations. It asks, given $k$ prescribed conjugacy classes %%%%1%%%%, whether there exist matrices $A_i \in C_i$ such that $A_1\cdots A_k = I$ and the tuple $(A_1, \dots, A_k)$ is irreducible, i.e., admits no common invariant subspace. The problem historically emerged from monodromy questions for Fuchsian differential equations on the Riemann sphere, but its modern formulation interweaves geometric moduli, quivers, and root theoretic criteria.

1. Problem Formulation and Historical Development

In its classical setting, the DSP is motivated by the monodromy of linear differential equations with regular singular points. It requires, for given conjugacy classes $C_1,\dots,C_k \subset GL_n(\mathbb{C})$, the existence of matrices $A_i \in C_i$ such that

$A_1A_2\cdots A_k = I$

and $\langle A_1, \dots, A_k \rangle$ acts irreducibly on $\mathbb{C}^n$. This connects to the realization of solutions of Fuchsian systems and more generally the construction of irreducible local systems with specified local monodromy. The additive and multiplicative variants (sum zero or product identity) encode compatibility constraints from global residue or monodromy relations.

The significance of the problem is in its role for classifying moduli spaces of connections (with fixed residues or local behavior), constructing flat bundles, and understanding symplectic geometry and integrable systems on these moduli spaces.

2. Quiver-Theoretic Reformulation and Root System Criterion

Recent advances reformulate the DSP via the theory of quiver representations. For given conjugacy data, one associates a star–shaped quiver $Q_w$ and a dimension vector $\alpha$ encoding the structure of flag varieties and ranks:

• Let $w = (w_1,\ldots,w_k)$ with $w_i$ the length of the flag at point $i$.
• The dimension vector $\alpha$ collects ranks $n_*$ and $n_{ij}$ from the prescribed conjugacy data, where $n_{ij}$ is the rank after applying the minimal polynomial factors for $C_i$.
• Associated are multiplicative parameters $\xi_{ij}$, the eigenvalues, forming the compatibility constraint $$\xi^{[\alpha]} := \prod_{i=1}^k\prod_{j=1}^{w_i} \xi_{ij}^{n_{i,j-1} - n_{ij}} = 1$$.

The central theorem confirmed in (Crawley-Boevey, 15 Sep 2025) states: There exists an irreducible solution to the DSP if and only if the dimension vector $\alpha$ is a positive root in the root system of $Q_w$, the multiplicative compatibility $\xi^{[\alpha]} = 1$ holds, and for any nontrivial decomposition $\alpha = \beta + \gamma + \cdots$ (with each $\xi^{[\beta]}=1$), the rigidity inequality

$p_w(\alpha) > p_w(\beta) + p_w(\gamma) + \cdots$

is satisfied, where $p_w$ is defined by the Euler form $q_w$ of the quiver via $p_w(\alpha) = 1 - q_w(\alpha)$.

3. Moduli Spaces, Parabolic Bundles, and Connection with Geometry

The geometric approach interprets the problem in terms of moduli spaces of parabolic bundles and connections (see (Soibelman, 2013)). Weighted projective lines and their categories of parabolic bundles (as defined by Geigle-Lenzing) provide a classification framework. A parabolic bundle is a vector bundle equipped with "flags" at marked points, and its dimension data precisely determine the quiver representation space.

When the moduli stack of parabolic bundles (with appropriate flags, weights, and dimension vector) satisfies the "very good" property, further geometric structures—such as symplectic structures and complete intersections—are available, enabling explicit calculation of existence, irreducibility, and dimension of solution spaces to DSP constraints.

Quiver varieties appear as GIT quotients, with moduli spaces of parabolic Higgs bundles exhibiting algebraically completely integrable system (ACIS) structures, as established in (Wen, 2021). Through isomorphisms between moduli of parabolic Higgs bundles and quiver varieties, solutions to the DSP (especially in nilpotent cases) acquire explicit geometric realization.

4. Connections with Differential Equations and Rigid Local Systems

The DSP is fundamentally tied to the existence of global connections matching prescribed local formal types at singularities. In the unramified or regular singular context, the problem is solved via quiver variety criteria (Hiroe, 2013). In more generality (additive or irregular versions), solution sets are characterized using spectral data, compatibility conditions, and rigidity indices (see the deformation and unfolding treatments of (Hiroe, 30 Jul 2024)).

Cohomologically rigid local systems—those whose moduli have vanishing tangent space—play a key role, as shown in (Esnault et al., 2017). Such systems are isolated points in the moduli stack, corresponding exactly to "rigid tuples" in the DSP. The arithmetic significance (integrality, companion sheaves, and finiteness of possibilities) further constrains the solution landscape and relates the DSP to Simpson's conjectures on geometric origin.

For connections with maximally ramified singularities (the Coxeter case), existence and rigidity are governed by trace and Jordan block count conditions (Kulkarni et al., 2021, Livesay et al., 2023). Explicit construction algorithms deploy techniques from matrix completion and graph theory to realize connections with designated local monodromy.

5. The Role of Nonabelian Hodge Correspondence

The nonabelian Hodge correspondence provides a categorical equivalence (moduli-level homeomorphism) between filtered local systems (Betti moduli) and parabolic Higgs bundles (Dolbeault moduli). This is fundamental for transferring existence questions in the DSP from character varieties to moduli of Higgs bundles with prescribed residue conditions (Hu et al., 2019, Wen, 2021, Lee et al., 10 Sep 2025). Stability conditions, spectral data, and the Hitchin fibration all play essential technical roles.

The correspondence allows the translation of analytic monodromy problems (flat connections with prescribed singularities) into more algebraically tractable moduli problems (parabolic Higgs bundles satisfying spectral conditions and $OK$ inequalities), thereby facilitating the geometric and combinatorial solution strategies, such as the $OK$ condition established in (Lee et al., 10 Sep 2025).

6. Root System Association and Multiplicative Preprojective Algebras

The translation of the DSP to quiver theory leads naturally to the paper of (multiplicative) preprojective algebras $\Lambda^q(Q)$, as the existence of an irreducible solution is equivalent to the existence of a simple module with dimension vector $\alpha$, as specified by the eigenvalue data (see (Crawley-Boevey, 15 Sep 2025)).

• The algebraic structure encodes the compatibility relation on eigenvalues and ensures indecomposability via the rigidity inequality $p_w(\alpha) > \sum p_w(\beta_i)$.
• The root system not only categorifies the problem but also enables applications of Kac–Moody techniques, reflection functors, and dimension counts.

This synthesis of ideas offers a paradigm for solving global–local problems across differential equations, representation theory, and algebraic geometry.

7. Implications, Classification, and Applications

The solution of the DSP in full generality—proven via root system criteria and rigidity inequalities—yields precise answers for the existence of irreducible matrix tuples, Fuchsian systems, and corresponding flat connections. This enables:

• Analytic classification of Fuchsian systems with prescribed local monodromies;
• Identification and construction of rigid local systems relevant to geometric Langlands theory;
• Algorithmic realization of connections with ramified singularities;
• Characterization of moduli spaces (non-emptiness, irreducibility, connectedness) via quiver varieties;
• Reduction of motivic questions (such as those arising in arithmetic geometry) to crystalline extension properties and the arithmetic of rank–2 local systems (Krishnamoorthy et al., 2018).

The framework constructed exposes deep interplay between global differential equations, root-theoretic combinatorics, quiver/algebraic geometry, and arithmetic integrality. Through moduli geometry and nonabelian Hodge theory, the DSP becomes an archetype for the interconnection of analysis, algebra, and geometry in modern mathematics.