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A Counterexample to Wehlau's Conjecture on Noether Numbers

This paper disproves Wehlau's conjecture that the Noether number of a $G$ -submodule is bounded by that of the ambient module by constructing a specific counterexample in characteristic 2 involving the dihedral group $D_8$ , where the Noether number of a 5-dimensional submodule exceeds that of its 6-dimensional parent module.

Original authors: Muhammad Fazeel Anwar

Published 2026-07-22

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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 a vast, invisible library where every book is a mathematical recipe for symmetry. This library belongs to a branch of math called representation theory, which studies how groups of symmetries (like rotating a shape or swapping colors) interact with spaces of numbers. In this library, the "Noether number" is a special scorecard. It tells you the highest complexity level needed to write down the most important "invariant" recipes for a specific symmetry group. An invariant is a rule that stays exactly the same no matter how you twist or turn the system. Think of it like a secret code that survives a storm: if you spin a cube, the fact that it still has six faces is an invariant. Mathematicians have long wondered if making a system bigger (adding more ingredients to the recipe) always makes the secret code easier or at least no harder to crack. For decades, a respected mathematician named Wehlau guessed that the answer was "yes": if you have a small system and a bigger one containing it, the bigger one should never require a more complex code than the smaller one. It felt like a safe bet, a rule that seemed to hold true in almost every case they checked.

But in the world of math, "almost always" is not the same as "always." A team led by Muhammad Fazeel Anwar has just found a very specific, very stubborn exception to this rule. They didn't just guess; they built a concrete, working model that proves the rule is broken. They focused on a group called the dihedral group D8 (the symmetries of a square) working in a world where the only numbers are 0 and 1 (a "characteristic 2" universe). They constructed two systems: a smaller one with 5 dimensions and a larger one with 6 dimensions. According to Wehlau's conjecture, the larger system should have a Noether number (complexity score) of 5 or less. Instead, the authors proved that the larger system has a score of 5, while the smaller system inside it has a score of 6. The smaller box is actually harder to crack than the bigger box that contains it.

This discovery is a definitive "no" to Wehlau's conjecture. The authors didn't just simulate this on a computer; they provided a rigorous, step-by-step mathematical proof that the conjecture fails in this specific setting. They showed that the complexity of these symmetry codes isn't just about the ingredients you have, but about how those ingredients are glued together. In fact, they proved that if you take a "split" version of their system (where the pieces are loosely connected) versus a "non-split" version (where they are tightly, messily knotted), the complexity changes completely, even though the basic building blocks are identical. This means that simply knowing the parts of a system isn't enough to predict how hard its secrets are to solve; the way the parts are woven together matters just as much. Their work doesn't just break a rule; it reveals that the landscape of symmetry is far more twisted and surprising than anyone previously thought.

Technical Summary: A Counterexample to Wehlau's Conjecture on Noether Numbers

Problem Statement
The paper addresses a conjecture proposed by Wehlau regarding the monotonicity of Noether numbers in the context of modular invariant theory. Let $G$ be a finite group acting on a finite-dimensional vector space $V$ over a field $k$ . The Noether number, denoted $\beta(V)$ , is defined as the largest degree of a homogeneous indecomposable element in the ring of invariants $k[V]^G$ . Wehlau conjectured that for any $G$ -submodule $U \subseteq V$ , the inequality $\beta(U) \leq \beta(V)$ holds. While this conjecture has been proven for specific cases (e.g., cyclic groups of prime order, certain dihedral groups in specific characteristics), its validity in the general modular case remained an open question.

Methodology
The authors construct a specific counterexample using the dihedral group $D_8$ over the field $k = \mathbb{F}_2$ . The methodology proceeds as follows:

Representation Construction:
- A 6-dimensional representation $V$ is defined via its dual space $X = V^*$ with basis $\{z_0, \dots, z_5\}$ . The action of the generators $r$ and $s$ of $D_8$ is explicitly defined on this basis.
- A 5-dimensional submodule $U \subset V$ is constructed by considering the quotient $Q = X / kz_0$ (where $kz_0$ is the fixed line under $G$ ) and taking its dual, $U = Q^*$ . This yields an injective $G$ -homomorphism $U \hookrightarrow V$ .
- The construction is defined over $\mathbb{F}_2$ but is noted to remain valid over any field of characteristic 2 via extension of scalars.
Computational Invariant Theory:
- The authors employ the algorithm described in [10] and the computer algebra system Magma [1] to compute the rings of invariants $k[V]^G$ and $k[U]^G$ .
- For $V$ : The invariant ring $R = \mathbb{F}_2[z_0, \dots, z_5]^G$ is analyzed. The authors identify a set of 11 homogeneous invariants $\{f_1, \dots, f_{11}\}$ and prove they form a minimal generating set. This is achieved by verifying that the subalgebra generated by these invariants matches the dimension of the full invariant space up to degree 12, utilizing Symonds' bound and a homogeneous system of parameters.
- For $U$ : The invariant ring $S = \mathbb{F}_2[x_0, \dots, x_4]^G$ is similarly analyzed. The authors identify a set of 9 homogeneous invariants $\{f_1, \dots, f_9\}$ and prove they form a minimal generating set using analogous dimension counting and Gaussian elimination techniques.

Key Results
The primary result is the explicit calculation of the Noether numbers for the constructed pair $(U, V)$ :

For the 6-dimensional module $V$ , the largest degree in the minimal generating set of $k[V]^G$ is 5. Thus, $\beta(k[V]^G) = 5$ .
For the 5-dimensional submodule $U$ , the largest degree in the minimal generating set of $k[U]^G$ is 6. Thus, $\beta(k[U]^G) = 6$ .

Consequently, the inequality $\beta(U) \leq \beta(V)$ is violated, as $6 > 5$ .

Significance and Consequences
The paper claims the following significance based on the counterexample:

Disproof of the Conjecture: The result definitively disproves Wehlau's monotonicity conjecture in the general modular case.
Generalization via Inflation: By inflating the $D_8$ -action through any surjection $\pi: H \twoheadrightarrow D_8$ , the counterexample extends to any finite group $H$ that admits $D_8$ as a quotient. This includes direct products $D_8 \times L$ , dihedral 2-groups of order $\geq 8$ , and generalized quaternion groups of order $\geq 16$ .
Structural Sensitivity: The example demonstrates that the modular Noether number is not determined solely by the Jordan–Hölder factors, the semisimplification, or the associated graded module. Specifically, the non-split extension $0 \to kz_0 \to X \to Q \to 0$ yields a lower Noether number ( $\beta=5$ ) than the split extension $Q \oplus k$ ( $\beta=6$ ), despite both having identical composition factors.
Generalized Noether Numbers: The paper notes that the same pair $(U, V)$ reverses the inequality for the first four generalized Noether numbers $\beta_k(G, W)$ , showing $\beta_k(G, U) > \beta_k(G, V)$ for $k=1, 2, 3, 4$ .

The work establishes that monotonicity of Noether numbers cannot be assumed in modular representation theory, even when the submodule has strictly lower dimension than the ambient module.

Technical Summary: A Counterexample to Wehlau's Conjecture on Noether Numbers

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