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Counterexamples to Norm Conjectures of Wehlau in Modular Invariant Theory

This paper constructs explicit counterexamples in characteristic 2 using faithful representations of elementary abelian 2-groups over $\mathbb{F}_8$ to disprove Wehlau's conjectures regarding the indecomposability of nonlinear orbit norms, including cases where the invariant ring is polynomial.

Original authors: Muhammad Fazeel Anwar

Published 2026-07-28

📖 8 min read🧠 Deep dive

CC BY 4.0

Original authors: Muhammad Fazeel Anwar

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 a master architect trying to build a fortress out of mathematical bricks. In this world, the bricks are simple shapes called "polynomials," and the fortress is a structure called an "invariant ring." This ring is special because it stays exactly the same even when you shake the ground beneath it. The shaking is done by a group of dancers (a "group") who spin and swap the bricks around. If the fortress looks identical after the dance, the bricks inside are "invariant."

For a long time, mathematicians have been trying to figure out the most efficient way to build these fortresses. They want to know: What is the smallest set of unique bricks needed to build the whole thing without any leftovers? In the 1990s, a mathematician named Wehlau made a bold guess about a specific type of brick called a "norm." He thought that if you took a single brick and spun it around the dance floor, the product of all its positions (the "orbit norm") would almost always be a fundamental, indestructible brick needed to build the fortress. It was a beautiful, tidy theory that promised to simplify how we understand these mathematical structures. But in the world of math, a theory is only as good as its ability to survive a crash test.

This paper is that crash test. The author, Muhammad Fazeel Anwar, sets out to see if Wehlau's guess holds up when the rules of the game get a little weird—specifically, when the math is done in "characteristic 2," a strange universe where adding two of the same thing equals zero (like 1 + 1 = 0). Anwar doesn't just test the theory; he builds a machine designed to break it. He constructs specific, four-dimensional dance floors where the dancers move in very precise ways. When he runs the experiment, he finds that Wehlau's "indestructible" norm bricks are actually just piles of smaller, already-used bricks glued together. In fact, he proves that in these specific cases, you can never use these norm bricks as the foundation of your fortress. He doesn't just suggest this might be true; he builds the exact mathematical counterexamples that prove the theory is false, showing that even when the fortress is perfectly built (a "polynomial" ring), the rules Wehlau proposed simply don't work.

The Story of the Broken Rule

Let's dive into the details of this mathematical heist. The paper focuses on a field of math called Modular Invariant Theory. Think of it as the study of patterns that survive chaos. Imagine you have a bag of colored marbles (the "vector space"), and a group of friends (the "group") keeps swapping them around. Some arrangements of marbles look the same no matter how the friends shuffle them. These unshakable arrangements are the "invariants."

The paper asks a very specific question: If you take one single marble and spin it around the room, creating a "norm" (which is just the product of all the places that marble visited), is that norm a special, unique building block? Or is it just a copy of blocks we already have?

Wehlau's conjecture was the "Norm Conjecture." It claimed that for most groups, these orbit norms are special. They are the unique, non-reducible bricks that you absolutely need to build the invariant ring. It was a comforting idea: a rule that said, "If you spin a marble, the result is always a new, essential piece of the puzzle."

Anwar, however, decided to test this in a very specific, tricky environment: Characteristic 2. In this mathematical universe, the number 2 doesn't exist; it's just 0. This changes how the "dancers" move. The author constructs two main scenarios to break the rule.

The First Heist: The $C_2^3$ Dance
First, Anwar sets up a dance floor with four dimensions (think of it as a hyper-4D room) and a group of eight dancers (specifically, a group called $C_2^3$ ). He chooses a field of numbers called $\mathbb{F}_8$ , which is like a tiny, finite universe of numbers. He defines exactly how the dancers move: they slide the marbles around in very specific patterns involving a special number $\alpha$ (where $\alpha^3 = \alpha + 1$ ).

He calculates the "invariant ring" for this setup. He finds that the fortress can be built with just five types of bricks, with sizes (degrees) of 1, 1, 4, 4, and 6. This is the "minimal" set; you can't build it with fewer or smaller bricks.

Then, he looks at the "orbit norms." He spins every possible non-fixed marble around the dance floor. Because the dancers move in a specific way, every spin creates a norm that is a product of 8 marbles. So, the size of every orbit norm is 8.

Here is the punchline: The fortress only needs bricks of size 1, 4, and 6. It has no need for a brick of size 8. Why? Because in this specific math world, a brick of size 8 is just a combination of the smaller bricks (specifically, it's in the square of the "positive degree" part of the ring). Anwar proves that every nonlinear orbit norm is "decomposable," meaning it's just a pile of smaller bricks glued together. It is not a unique, essential building block. This proves that Wehlau's idea that these norms are special is false.

The Second Heist: The $C_2^4$ Perfect Fortress
To make the point even stronger, Anwar builds a second scenario. This time, he uses a group of 16 dancers ( $C_2^4$ ) on the same 4D floor. He sets up the moves so that the resulting invariant ring is a "polynomial algebra." In math-speak, this is the "perfect" fortress: it's built cleanly with no messy overlaps, just like a perfect tower of blocks.

He finds that this perfect fortress needs bricks of size 1, 1, 4, and 4. Again, he spins the marbles. The resulting orbit norms have sizes of either 8 or 16.

If the norm is size 8, it's bigger than the 4-sized bricks needed.
If the norm is size 16, it's even bigger.

Anwar shows that in this perfect, polynomial fortress, every orbit norm is still just a pile of smaller bricks. Even though the fortress is perfectly built, the "norm" bricks are useless as foundations. This disproves a stronger version of Wehlau's conjecture, which claimed that if the fortress is a polynomial ring, the norms must be essential. Anwar says: "Not so fast. Here is a perfect fortress, and the norms are still just junk."

The Third Twist: The Full Group Norm
Finally, the paper tackles a slightly different version of the rule: the "full group norm." This is where you multiply the marble by every possible position the group can put it in, not just the unique spots.

Anwar constructs a simple 3D example with a group of four dancers ( $C_2 \times C_2$ ). He shows that for this group, the invariant ring is also a perfect polynomial ring. However, the bricks needed to build this fortress have sizes of 1, 2, and 2 (specifically, one brick of size 1, and two bricks of size 2).

When he calculates the full group norm for any marble, he finds that the result is always a square of something else. In math terms, it's "decomposable."

He explains that this happens because every marble has a "stabilizer"—a part of the dance group that doesn't move it. Because of this, the full norm ends up being the same thing multiplied by itself, making it a "square." And in this world, squares are never unique, essential bricks; they are always just copies of existing things.

The Takeaway

The paper concludes with a definitive "No." Wehlau's conjectures, which suggested that orbit norms are the secret sauce for building these mathematical fortresses, are false. Anwar didn't just find a small exception; he built entire worlds where the rule completely collapses.

He proved that:

You can have a complex fortress where the orbit norms are never the essential bricks.
You can have a perfect polynomial fortress where the orbit norms are still never the essential bricks.
You can have a perfect fortress where the full group norms are never the essential bricks.

The paper doesn't just suggest this might be true; it provides explicit, calculated examples with specific numbers (like degrees 1, 1, 4, 4, 6) and specific groups ( $C_2^3$ , $C_2^4$ ) that serve as undeniable proof. The "Norm Conjecture" is dead, at least in the world of characteristic 2. The lesson for the curious teenager? In math, even the most elegant rules can crumble when you look at them through the right (or wrong) lens. Sometimes, the thing you think is a unique building block is just a pile of rubble.

Technical Summary: Counterexamples to Norm Conjectures of Wehlau in Modular Invariant Theory

Problem Statement
The paper addresses specific conjectures proposed by Wehlau regarding the role of norms in modular invariant theory. Let $G$ be a finite group acting on a finite-dimensional vector space $V$ over a field $k$ of characteristic $p$ dividing $|G|$ (the modular case). The ring of invariants is denoted $k[V]^G$ . The paper investigates two types of norms associated with a linear form $\ell \in V^*$ :

The orbit norm: $N^{\text{orb}}_G(\ell) = \prod_{m \in G\ell} m(\ell)$ , where the product runs over the distinct elements of the orbit.
The full group norm: $N^{\text{full}}_G(\ell) = \prod_{g \in G} g(\ell)$ .

Wehlau's conjectures posited that under certain conditions (specifically when the invariant ring is polynomial or when orbits are regular), these norms should appear as minimal algebra generators of the invariant ring. The paper seeks to determine if nonlinear orbit norms and full group norms can serve as indecomposable elements in a minimal homogeneous generating set for $k[V]^G$ .

Methodology
The author constructs explicit counterexamples using faithful representations of elementary abelian $p$ -groups ( $C_2^3$ and $C_2^4$ ) over finite fields of characteristic 2. The methodology involves:

Defining Specific Actions: Constructing faithful four-dimensional representations of $C_2^3$ and $C_2^4$ over the field $\mathbb{F}_8$ (and a three-dimensional representation of $C_2^2$ over $\mathbb{F}_2$ ). The group actions are defined by fixing certain coordinates and applying affine translations to others via specific shift matrices.
Computing Invariant Rings: Explicitly calculating the generators of the invariant rings $k[V]^G$ for these representations. This involves verifying that specific polynomials are invariant, establishing that they form a homogeneous system of parameters, and proving that the ring is generated by these polynomials (often by comparing Hilbert series or using integrality arguments).
Analyzing Orbit Structures: Determining the stabilizers and orbit sizes of non-fixed linear forms under the group action.
Degree Analysis: Comparing the degrees of the orbit norms and full group norms against the degrees of the minimal generators of the invariant ring. The core logic relies on the definition of a decomposable element: an element is decomposable if it lies in the square of the ideal of positive-degree elements, $(R_+)^2$ . If the degree of a norm exceeds the degrees of the minimal generators in a way that forces it into $(R_+)^2$ , it cannot be a minimal generator.

Key Contributions and Results

Disproof of the Unrestricted Norm Conjecture (Theorem 1.1):
The paper constructs a faithful representation of $G \cong C_2^3$ over $\mathbb{F}_8$ on $S = k[x_1, x_2, y_1, y_2]$ .
- Result: The invariant ring $S^G$ is minimally generated by homogeneous elements of degrees $\{1, 1, 4, 4, 6\}$ .
- Norm Analysis: Every non-fixed linear form has an orbit of size 8. Consequently, every nonlinear orbit norm has degree 8.
- Conclusion: Since the maximal degree of a minimal generator is 6, the degree 8 orbit norms must lie in $(S^G_+)^2$ . Thus, no nonlinear orbit norm can be a minimal generator, disproving the unrestricted form of Wehlau's conjecture.
Disproof of the Polynomial-Ring Norm Conjecture (Theorem 1.2):
The paper constructs a faithful representation of $G \cong C_2^4$ over $\mathbb{F}_8$ on $S = k[x_1, x_2, y_1, y_2]$ .
- Result: The invariant ring $S^G$ is a polynomial algebra with minimal generators of degrees $\{1, 1, 4, 4\}$ .
- Norm Analysis: Non-fixed linear forms have orbit sizes of either 8 or 16. Thus, nonlinear orbit norms have degrees 8 or 16.
- Conclusion: Even in the case where the invariant ring is polynomial, the orbit norms (degrees 8, 16) are strictly higher than the maximal generator degree (4). They are therefore decomposable. This disproves Wehlau's conjecture specifically for polynomial invariant rings.
Disproof for Full Group Norms (Theorem 1.3):
The paper presents a representation of $G \cong C_2 \times C_2$ over $\mathbb{F}_2$ on $R = k[x, y, z]$ .
- Result: The invariant ring $R^G$ is a polynomial algebra generated by $\{x, y^2+xy, z^2+xz\}$ .
- Norm Analysis: For every non-zero linear form $\ell$ , the stabilizer in $G$ is non-trivial (size $\ge 2$ ). The full group norm $N^{\text{full}}_G(\ell)$ is the product of the orbit elements raised to the power of the stabilizer size.
- Conclusion: Because the stabilizer size is at least 2, the full group norm is a square of an element in $R^G_+$ , placing it in $(R^G_+)^2$ . Thus, no full group norm of a linear form can be a minimal generator.

Significance and Claims
The paper claims to provide definitive counterexamples to both Wehlau's norm conjecture and its stronger, unrestricted extension.

Refutation of Sufficiency: The results demonstrate that the existence of a regular orbit (where the stabilizer is trivial) is not sufficient to guarantee that an orbit norm appears in a minimal generating set (as shown in Theorem 1.1).
Polynomial Rings: The paper establishes that even when the invariant ring is a polynomial algebra—a condition often associated with "nice" behavior in invariant theory—the norm conjecture fails (Theorem 1.2).
Distinction of Norms: The work highlights the essential distinction between orbit norms and full group norms. While orbit norms can sometimes be minimal generators (as seen in the standard Klein four-group examples cited in the introduction), the full group norm is shown to be inherently decomposable whenever the action on linear forms lacks a regular orbit (Remark 4.2).

The paper concludes that the conjectures regarding norms as minimal generators are false in the modular setting, even under restrictive conditions such as polynomial invariant rings or regular orbits.

The Story of the Broken Rule

The Takeaway

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