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$e$ -polynomials of character varieties

This paper presents lecture notes from a 2025 workshop that introduces $e$ -polynomials and mixed Hodge structures for character varieties, detailing arithmetic and geometric techniques to compute them, providing a complete example for the ${\rm GL}_3$ -character variety of a free group, and extending these methods to general reductive groups to derive motivic expressions and address topological mirror symmetry conjectures.

Original authors: Alfonso Zamora

Published 2026-06-26

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alfonso Zamora

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 understand the shape of a very strange, complex building. This building isn't made of bricks and mortar, but of mathematical possibilities. Specifically, it's a "Character Variety."

Think of a Character Variety as a giant map of all the possible ways a group of instructions (like a set of rules for a game) can be played out using a specific set of tools (a matrix group like $GL_n$ ). Some of these maps are smooth and easy to walk on; others are jagged, full of holes, and incredibly twisted.

The author, Alfonso Zamora, is trying to measure these maps. But because they are so weird, you can't just use a ruler. You need a special kind of "mathematical scanner" called an e-polynomial.

Here is a breakdown of the paper's journey, using everyday analogies:

1. The Problem: Measuring the Unmeasurable

In the world of smooth, perfect shapes (like a sphere), mathematicians have a long history of measuring them. But Character Varieties are often "singular"—they have sharp corners, self-intersections, and tears.

The Analogy: Imagine trying to count the number of rooms in a house that keeps changing its shape, has rooms inside rooms, and some walls that don't exist.
The Solution: The paper uses Mixed Hodge Structures. Think of this as a sophisticated X-ray that doesn't just see the shape, but sees the "texture" and "history" of the shape. It breaks the complex object down into layers of information (weights) that tell us about its topology (how many holes it has) and its geometry (how it curves).

2. The Tool: The e-Polynomial

Once you have this X-ray, you can summarize the entire complex shape into a single, manageable formula called the e-polynomial.

The Analogy: If the Character Variety is a complex 3D sculpture, the e-polynomial is like a barcode. You can't see the whole sculpture just by looking at the barcode, but if you scan it, it tells you the "Euler characteristic" (a fundamental count of the object's features, like vertices minus edges plus faces).
Why it's cool: This barcode works even for the jagged, broken shapes that usually break other measuring tools.

3. The Two Ways to Scan the Building

The paper introduces two main ways to calculate this barcode:

A. The Arithmetic Method (Counting in Finite Worlds)

Instead of looking at the infinite, complex building, the author suggests looking at a "pixelated" version of it over a finite field (a world with a limited number of points, like a grid with only 100 dots).

The Analogy: Imagine you want to know how many people live in a massive city. Instead of counting everyone, you count how many people live in a tiny, specific neighborhood with a known rule. You find a pattern (a polynomial) that predicts the population based on the size of the neighborhood.
The Magic: The paper claims that if you find this pattern for the "pixelated" version, it is exactly the same as the e-polynomial for the real, complex version. It's like realizing the blueprint for a tiny model is identical to the blueprint for the skyscraper.

B. The Geometric Method (The Lego Strategy)

The building is too big to measure all at once. So, the author breaks it down into smaller, manageable chunks called strata.

The Analogy: Imagine a messy pile of Legos. Instead of trying to describe the whole pile, you sort them by color and size.
- The "Irreducible" Stratum: These are the unique, one-of-a-kind Lego creations that can't be broken down further.
- The "Abelian" Stratum: These are the simple, repetitive patterns (like a straight line of identical bricks).
- The "Parabolic" Strata: These are the intermediate, complex structures.
The Strategy: The paper shows that you can calculate the barcode for each small pile of Legos separately and then just add them up to get the barcode for the whole mess. This is much easier than measuring the whole mess at once.

4. The Big Example: The Free Group

The author demonstrates this with a specific, difficult case: The Character Variety of a Free Group (a group with no rules other than the ones you make up) for $GL_3$ (3x3 matrices).

The Process:
1. They calculated the barcode for the "unique" Lego creations (irreducible representations).
2. They calculated the barcode for the "simple" Lego creations (abelian representations).
3. They calculated the "middle" ones.
4. They added them all up to get the final formula for the whole $GL_3$ variety.
The Result: They proved the resulting shape is "irreducible" (it's one single, connected piece) and calculated its exact Euler characteristic (which turned out to be zero).

5. The Grand Unification: Mirror Symmetry

Finally, the paper takes this Lego strategy and applies it to a much bigger question: Topological Mirror Symmetry.

The Concept: In physics and math, there are "twin" groups (called Langlands duals, like $SL_n$ and $PGL_n$ ) that look different but are supposed to have the same underlying "soul" or topology.
The Breakthrough: The paper argues that to prove these twins are identical, you don't need to compare the whole giant building. You just need to compare their Lego piles (the strata) one by one.
The Analogy: If you want to prove two different-looking houses are actually the same, you don't need to walk through every room. You just need to prove that every single room in House A has a matching room in House B. If every "stratum" matches, the whole buildings match.

Summary

This paper is a guidebook for measuring the unmeasurable. It teaches us how to:

Use arithmetic tricks (counting points in finite worlds) to understand infinite shapes.
Use geometric decomposition (breaking shapes into Lego-like layers) to simplify complex calculations.
Apply these methods to prove that mathematical twins (dual groups) share the same fundamental structure, piece by piece.

It turns a terrifyingly complex mathematical problem into a manageable, step-by-step construction project.

Technical Summary: e-Polynomials of Character Varieties

Problem Statement

The paper addresses the computation of topological and geometric invariants for character varieties, defined as the Geometric Invariant Theory (GIT) quotient $X_G(\Gamma) = \text{Hom}(\Gamma, G)//G$ , where $G$ is a complex reductive algebraic group and $\Gamma$ is a finitely presented group. While these spaces are well-understood when $\Gamma$ is the fundamental group of a Riemann surface (due to the non-abelian Hodge correspondence linking them to Higgs bundles), they become highly singular and difficult to analyze when $\Gamma$ is a free group or other general groups.

The primary objective is to compute the e-polynomial (a specialization of the mixed Hodge polynomial) for these character varieties. The e-polynomial encodes the mixed Hodge numbers of the cohomology with compact support, serving as a bridge between topology (Betti numbers) and complex geometry. A significant motivation for these computations is the investigation of topological mirror symmetry, a conjecture stating that character varieties associated with Langlands dual groups (e.g., $SL_n$ and $PGL_n$ , or $Sp_{2n}$ and $SO_{2n+1}$ ) possess equal or "mirrored" invariants.

Methodology

The manuscript synthesizes arithmetic and geometric techniques to compute e-polynomials, building upon foundational work by Hausel, Rodriguez-Villegas, Katz, and others.

1. Mixed Hodge Structures and e-Polynomials

The paper establishes the theoretical framework using Deligne's mixed Hodge structures for singular and non-compact varieties.

Definition: The e-polynomial $e(X; u, v)$ is defined as $\sum (-1)^k h^{k,p,q}_c(X) u^p v^q$ , where $h^{k,p,q}_c$ are the compactly supported mixed Hodge numbers.
Arithmetic Connection: Following Katz's appendix in [HR08], for varieties of "polynomial type," the number of points over a finite field $\mathbb{F}_q$ is given by a polynomial $C_X(q)$ which coincides with the e-polynomial evaluated at $x=uv$. This allows the use of point-counting over finite fields to determine topological invariants.
Properties: The e-polynomial is additive over locally closed stratifications and multiplicative under certain fibrations (specifically when the base acts trivially on the cohomology of the fiber).

2. Arithmetic Methods (Counting Points)

For free groups $\Gamma = F_r$ , the paper utilizes combinatorial methods involving Hall algebras and plethystic operators (inspired by [MR09; MR15]).

The number of irreducible representations over $\mathbb{F}_q$ is related to the total number of representations via the plethystic exponential ($PExp$) and plethystic logarithm ($PLog$).
The Adams operator $\Psi$ and its inverse are used to manipulate generating series of representation counts, allowing the derivation of explicit formulas for the e-polynomials of irreducible character varieties $X^*_G(F_r)$ .

3. Geometric Stratification

To handle the full character variety (including reducible representations), the paper employs a stratification by partition type (generalizing the Jordan type stratification).

Partition Stratification: The representation space is decomposed into strata $X^{[k]}_G(\Gamma)$ indexed by partitions $[k]$ of $n$ (for $GL_n$ ). Each stratum corresponds to representations that are polystable with a specific stabilizer structure (isomorphic to a direct sum of irreducible representations of smaller dimensions).
Reduction to Cores: A key geometric innovation involves the concept of pseudo-quotients and cores (from [Gon24]). The paper proves that the motivic class (and thus the e-polynomial) of a stratum $X^{[k]}_G(\Gamma)$ can be reduced to the quotient of a "core" subvariety by a finite group (the Weyl group of the Levi subgroup).
Generalization to Reductive Groups: This stratification is extended from $GL_n$ to general complex reductive groups $G$ using root data. The stratification is indexed by subsets of simple roots (parabolic subgroups), reducing the computation of $X_G(\Gamma)$ to a sum of e-polynomials of character varieties of Levi subgroups $L_I$ modulo their normalizers $N_I$ .

Key Contributions and Results

1. Explicit Computation for $GL_3$

The paper provides a complete, step-by-step calculation of the e-polynomial for the $GL_3$ -character variety of the free group $F_r$ .

Step 1: Computes the e-polynomials for the irreducible strata $X^*_{GL_1}, X^*_{GL_2}, X^*_{GL_3}$ using the arithmetic/plethystic method.
Step 2: Computes the e-polynomial for the abelian stratum (corresponding to the partition $[1^3]$ ), which is isomorphic to the character variety of the abelianization $\mathbb{Z}^r$ .
Step 3: Combines these using the stratification formula to derive the explicit polynomial for $X_{GL_3}(F_r)$ .
Result: The resulting polynomial confirms that the variety is irreducible (leading coefficient is 1) and has complex dimension $9r-8$ .

2. General Stratification for Reductive Groups

The manuscript generalizes the partition stratification to any complex reductive group $G$ (Types A, B, C, D).

It establishes a decomposition of the character variety $X_G(\Gamma)$ into strata indexed by parabolic subgroups.
It proves that the e-polynomial of the whole variety is the sum of the e-polynomials of these strata, where each stratum is expressed in terms of irreducible character varieties of Levi subgroups modulo finite Weyl groups.
This reduces the problem of computing invariants for high-rank groups to recursive computations involving lower-rank groups and finite group actions.

3. Topological Mirror Symmetry

The paper applies this stratification to the topological mirror symmetry conjecture for Langlands dual pairs.

Reduction: It demonstrates that proving $e(X_G(\Gamma)) = e(X_{^L G}(\Gamma))$ reduces to verifying the equality stratum-by-stratum.
Type A ( $SL_n$ vs $PGL_n$ ): The stratification recovers known results, confirming mirror symmetry for these groups.
Types B and C ( $Sp_{2n}$ vs $SO_{2n+1}$ ): The paper derives explicit decomposition formulas for both groups. It shows that mirror symmetry holds if and only if it holds for the irreducible components and specific mixed strata involving products of $GL$ and symplectic/orthogonal groups.
Example: For $Sp_6$ and $SO_7$ , the paper explicitly identifies the strata where the Levi subgroups differ ( $GL_1 \times Sp_4$ vs $GL_1 \times SO_5$ ) and notes that proving mirror symmetry for the full variety reduces to proving it for these specific lower-rank components.

Significance and Claims

The paper claims to provide a unified framework for computing e-polynomials of character varieties for arbitrary reductive groups, moving beyond the limitations of previous arithmetic-only or low-rank geometric methods.

Unification: It successfully combines arithmetic counting (effective for irreducible representations) with geometric stratification (necessary for handling singularities and reducible representations).
Computational Feasibility: By introducing the "core" and "pseudo-quotient" concepts, the paper reduces complex infinite-dimensional GIT quotients to manageable finite quotients of lower-dimensional varieties.
Mirror Symmetry: The work provides a concrete pathway to test topological mirror symmetry conjectures. Rather than attempting to compute the invariants of the entire moduli space directly, the problem is reduced to checking equality on specific, lower-dimensional strata. The paper explicitly states that while mirror symmetry is proven for Type A and certain cases of Type B/C, the general case for higher ranks remains an open problem, specifically requiring the verification of equality for irreducible components of dual groups.

The manuscript serves as both a technical guide for performing these calculations (including a full example for $GL_3$ ) and a theoretical extension of stratification techniques to general reductive groups, offering a structured approach to future research in the field.