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Dimension bounds for relative character varieties on the projective line with three punctures $G=GL(r), O(r), Sp(r)$

This paper establishes explicit linear upper bounds on the rank of MC-minimal relative character varieties for $GL(r)$, $O(r)$ , and $Sp(r)$ on the three-punctured projective line using Simpson's diagrammatic method, demonstrating that any such variety is isomorphic via Katz's middle convolution to one satisfying these bounds, which are shown to be sharp for the general linear and non-overlapping quadratic cases.

Original authors: Emmett Lennen

Published 2026-02-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Emmett Lennen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to build a very specific type of castle. This castle isn't made of bricks, but of mathematical shapes called "character varieties." These castles exist on a strange, magical landscape: a sphere (like the Earth) with three holes punched in it (at the North Pole, the South Pole, and the Equator).

The goal of this paper is to answer a simple but tricky question: "If I tell you how big the inside of your castle is (its dimension), how big can the foundation (the rank) possibly be?"

Here is the breakdown of the paper's journey, translated into everyday language.

1. The Setting: The Three-Hole Sphere

Imagine a balloon with three holes. You are drawing paths around these holes. Because there are holes, the paths can get "stuck" or twisted in different ways.

The Rank ( $r$ ): Think of this as the width of your foundation. It's how many layers of bricks you are using to build the base. A higher rank means a wider, more complex foundation.
The Dimension ( $d$ ): Think of this as the total floor space inside the castle. It's the amount of "room" you have to move around.
The Problem: Usually, if you want a huge floor space, you need a huge foundation. But sometimes, you can cheat the system and get a lot of space with a surprisingly small foundation. The author wants to find the absolute limit: "What is the biggest foundation you could possibly need for a given amount of floor space?"

2. The Magic Tool: Middle Convolution

The author uses a magical tool invented by a mathematician named Katz, called Middle Convolution.

The Analogy: Imagine you have a messy pile of Lego bricks (your foundation). You want to know if you can shrink the pile down without losing the shape of the castle.
The Process: Middle Convolution is like a "shrink-ray" for your foundation. It takes your complex foundation and tries to compress it.
MC-Minimal: If you use the shrink-ray and the foundation cannot get any smaller without breaking the castle, you have reached an MC-minimal state. This is the "most efficient" version of your castle. The paper only cares about these efficient versions because if you can shrink a big castle down to a small one, the big one isn't the limit case.

3. The Two Types of Castles

The paper looks at two different types of building materials (groups):

The General Linear Castles ($GL(r)$): These are the standard, flexible castles. They are like building with standard Lego bricks.
The Quadratic Castles ( $O(r)$ and $Sp(r)$): These are more rigid. They have to follow strict symmetry rules (like a reflection in a mirror or a rotation). They are like building with specialized, interlocking puzzle pieces.

4. The Diagrammatic Method: The "Box" Game

To solve the problem, the author (and the mathematician Simpson before him) uses a visual trick.

The Grid: Imagine a giant square grid of size $r \times r$ .
The Squares: You have to fit three sets of smaller squares into this big grid, one set for each of the three holes.
The Rule: The "Dimension" (floor space) is simply the empty white space left over after you place all your colored squares.
The Goal: To get a small amount of empty space (low dimension), you have to pack the colored squares very tightly. To get a huge foundation ( $r$ ) with a tiny amount of empty space, you have to pack them in a very specific, inefficient way.

5. The Breakthrough: Tightening the Bounds

Previous mathematicians (like Simpson) had found a rule for the quadratic castles, but it was a bit loose. It was like saying, "If your castle has 10 rooms, your foundation might be up to 1,000 bricks wide." That's a huge range!

Emmett Lennen's contribution:
He went back to the drawing board and tightened the rules.

For the Standard Castles ($GL$): He proved that if your castle has dimension $d$ , your foundation can never be wider than $3d$ . (If you have 10 rooms, your foundation is at most 30 bricks wide).
For the Rigid Castles (Quadratic): He improved the previous massive number to a much tighter bound: $9d + 54$ .

6. How Did He Do It? (The Detective Work)

The author didn't just guess. He treated the problem like a detective solving a puzzle with thousands of suspects (possible configurations of squares).

The "Non-Overlapping" Case: First, he looked at the easy cases where the three sets of squares didn't bump into each other. He proved that even in the worst-case scenario, the foundation couldn't be too big.
The "Overlapping" Case: Then, he looked at the tricky cases where the squares did overlap. He showed that any overlapping castle could be "un-overlapped" into a slightly smaller version, which meant he could use his previous results to solve this harder case too.
The "Dominance Ordering": This is a fancy way of saying, "If you move a square from a crowded spot to an empty spot, you usually make the castle less efficient (bigger dimension)." He used this logic to eliminate thousands of impossible scenarios, leaving only a few "suspects" to check manually.

The Bottom Line

This paper is a mathematical "speed limit" sign.

Before: We knew that if you wanted a certain amount of space, your foundation couldn't be infinitely big, but the limit was vague and huge.
Now: We know the exact speed limit. If you want a castle of a certain size, the foundation cannot exceed a specific, much smaller number.

Why does this matter?
In mathematics, knowing the limits helps us understand the fundamental structure of these shapes. It tells us that these complex mathematical objects are much more "tightly packed" and efficient than we previously thought. It's like realizing that a massive, complex machine can actually be built with far fewer gears than we thought possible.

1. Problem Statement

The paper investigates the relationship between the rank ( $r$ ) and the dimension ( $d$ ) of relative character varieties on the punctured projective line $\mathbb{P}^1 \setminus \{0, 1, \infty\}$ . Specifically, it focuses on the groups $G = GL(r)$, $O(r)$ , and $Sp(r)$.

Context: A relative character variety parametrizes irreducible representations of the fundamental group of the punctured sphere into $G$ , with fixed conjugacy classes for the monodromy around the punctures.
Goal: To establish an explicit linear upper bound $R(d)$ on the rank $r$ for an MC-minimal character variety of a given dimension $d > 2$ .
MC-Minimality: A character variety is "MC-minimal" if the operation of middle convolution (introduced by Katz) cannot further reduce its rank. Katz's algorithm implies that any character variety is isomorphic (via middle convolution) to an MC-minimal one. Understanding the bounds for these minimal cases is crucial for classifying all such varieties.
Specific Challenge: While Simpson [Sim06] provided a diagrammatic method for $GL(r)$, the quadratic case ( $O(r)$ and $Sp(r)$) is more complex due to the existence of different eigenvalue types (paired, $+1$ , $-1$) and the need for higher-rank convolution kernels. Previous bounds for the quadratic case (derived from [Sim24]) were extremely loose (e.g., $O(10^{10} d)$ ).

2. Methodology

The author employs a combinatorial and diagrammatic approach based on Simpson's work, analyzing the "box dimension" of configurations associated with the character varieties.

A. Combinatorial Setup

The problem is translated into analyzing partitions of the rank $r$ associated with the monodromy eigenvalues at the three punctures ( $t_1, t_2, t_3$ ).

GL(r) Case: The rank $r$ is split into three columns of widths $q_1, q_2, q_3$ (where $\sum q_i = r$ ). The dimension is calculated as the "empty area" in an $r \times r$ square after removing squares corresponding to the eigenvalue multiplicities.
Quadratic Case ($O(r)/Sp(r)$): The setup is similar but involves three types of eigenvalues:
1. Type $m$ : Paired eigenvalues $z, z^{-1}$ .
2. Type $e$ : Eigenvalue $1$ (or $-1$ depending on parity).
3. Type $f$ : Eigenvalue $-1$ (or $1$).
  The dimension formula includes a "box dimension" ( $\beta$ ) and a "linear correction" ( $l$ ) term: $\Delta = \beta - \epsilon l$ .

B. Reduction Strategy

The core of the proof involves reducing the infinite space of possible partitions to a finite set of "critical" cases:

Dominance Ordering: The author utilizes the dominance ordering of partitions. A key lemma (Prop 4.5) proves that moving a partition "down" in the dominance order (making it less uniform) strictly increases the dimension. Thus, the minimal dimension for a fixed rank occurs at the "most uniform" partitions (e.g., $(q, q, \dots, k)$ ).
Parameterization: The rank is written as $r = n_i q_i + c_i$ , where $q_i$ is the column width and $c_i$ is a remainder term measuring how close $q_i$ is to dividing $r$ .
Case Reduction: By analyzing the inequalities derived from the dimension formulas, the author proves that if the dimension is small relative to the rank (specifically $\Delta \leq r/3 - 12$ ), the parameters $(n_1, n_2)$ must belong to a finite set: $\{(2,2), (3,3), (2,4), (2,3)\}$ , and $|c_i| \leq n_i$ .
Exhaustive Check: The remaining finite cases are checked computationally (Section 5) to find the exact configurations that minimize the dimension.

3. Key Contributions and Results

A. General Linear Case ($GL(r)$)

Result: The paper confirms and refines Simpson's result, proving that for any MC-minimal character variety of dimension $d > 2$ , the rank satisfies:
$r \leq 3d$
Sharpness: The bound is sharp. Equality holds for the specific configuration where $(q_1, q_2, q_3) = (r/2, r/3, r/6)$ with specific column partitions.

B. Quadratic Case ($O(r), Sp(r)$)

This is the primary novel contribution of the paper.

Previous Bounds: Simpson's proof in [Sim24] implied a bound of roughly $3/2(1260d + \text{huge constant})$ .
New Bound: The author proves a significantly tighter linear bound:
$r \leq 9d + 54$
(Specifically, for non-overlapping configurations, $r \leq 6d + 36$ ; the factor of $3/2$ accounts for overlapping configurations).
Methodological Improvement: Instead of relying on Simpson's complex reductions, the author introduces a refined parameterization ( $r = n_i q_i + c_i$ ) and performs a rigorous case-by-case analysis of the dominance ordering to eliminate large classes of partitions efficiently.
Sharpness: The bound is achieved by specific configurations described in Example 4.8 (involving partitions like $(r/2, r/2)$ , $(r/3, r/3, r/3)$ , and $(r/6, \dots)$ with specific type assignments).

C. Non-Overlapping vs. Overlapping

The paper distinguishes between:

Non-overlapping: The squares representing eigenvalues fit into disjoint rectangles.
Overlapping: Occurs in the quadratic case when specific eigenvalues overlap.
The author proves that any overlapping configuration of dimension $d$ has a rank bounded by $3/2$ times the bound of the corresponding non-overlapping configuration, allowing the focus to remain on the non-overlapping case for the primary bound derivation.

4. Significance

Tightening Bounds: The paper drastically improves the known upper bounds for the rank of quadratic character varieties, reducing the coefficient of $d$ from $\approx 1890$ (in the worst interpretation of [Sim24]) to $9$. This makes the classification of these varieties much more tractable.
Algorithmic Implications: Since middle convolution is the primary tool for constructing and classifying rigid local systems (dimension 0) and their deformations, knowing the precise bounds on the rank of minimal varieties helps in determining the complexity of algorithms searching for such systems.
Combinatorial Rigor: The paper provides a robust framework for analyzing dimension bounds using dominance ordering and column partition analysis, which could be applicable to other problems involving moduli spaces of local systems.
Addressing the Deligne-Simpson Problem: While the paper focuses on dimension bounds (assuming non-emptiness), it explicitly discusses the Deligne-Simpson problem (conditions for the existence of such varieties). The results provide the necessary dimensional constraints that any solution to the Deligne-Simpson problem must satisfy.

5. Conclusion

Emmett Lennen's paper successfully establishes the sharpest possible linear upper bounds for the rank of MC-minimal character varieties on $\mathbb{P}^1$ with three punctures. By refining Simpson's diagrammatic methods and introducing a rigorous case-analysis based on partition dominance, the author reduces the quadratic bound to $r \leq 9d + 54$ , a significant improvement over previous estimates. This work solidifies the structural understanding of relative character varieties for classical groups and provides a clear path for future investigations into the existence (non-emptiness) of these varieties.