The Mathieu conjecture for SU(2)
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Mathieu’s SU(2) conjecture asks whether eventual vanishing of all pure Haar moments of a regular function forces eventual vanishing after multiplication by any fixed regular function.
Why it matters and what counts as progress
Why it matters: The SU(2) case is a concrete compact-group test not settled merely by failure of a broader universal conjecture.
What counts as progress: A proof of the implication or explicit functions F,G with vanishing pure moments and persistently nonzero marked moments, with the SU(2) lift justified.
How the problem developed
- 1997 · Conjecture [1]
Olivier Mathieu Proposed the compact-group vanishing conjecture that moments vanishing eventually force the same eventual vanishing after multiplication by any fixed regular function.
- 21 Jul 2026 · Counterexample [2]
Christopher D. Long Gave an explicit Laurent-polynomial counterexample that lifts to regular functions on SU(2), disproving the Mathieu conjecture for this group and all mixed xz cases by padding.
AI contribution
Counterexamples to the xz-Conjecture and the Mathieu Conjecture for SU(2)
The Laurent polynomial f(x,z)=(1−z⁻¹)((1−x)+xz) has zero pure product-integral moments for every positive n but marked moment (−1)^(n−1)/(n+1). It refutes xz(1,1), with padding for all mixed k,l≥1. Its SU(2) lift F=(1+c)(ad+b), G=−c has the same pure and marked Haar moments and refutes the SU(2) Mathieu conjecture.
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AI-found polynomials refute the mixed xz and SU(2) Mathieu conjectures