Uniform irreducibility analysis for all even levels

Determine whether the leading-character representation and the full two-point representation associated with the extremal affine \(sl(2)\) family are irreducible for every even level k, and develop a systematic treatment beyond the subset-product criterion.

Background

The paper proves irreducibility of the three-dimensional leading representation at level k=2 by checking the subset-product criterion of KSW. For general even k, the number of subset products grows exponentially, and failure of the criterion would not itself imply reducibility. The paper therefore leaves unresolved both a uniform irreducibility analysis and the corresponding question for the full, sub-leading two-point representation.

References

For general even k, the analogous check involves 2{k+1}-2 subset sums of the exponents \bigl(2\mu2+4\mu-k\bigr)/\bigl(8(k+2)\bigr), \mu=0,\dots,k, and we do not know a uniform argument (of the type used in KSW Theorem 5.6 for the non-congruence question) settling irreducibility for all k.

— Modular properties of affine \(\SLA{sl}{2}\) torus \(n\)-point functions  (2609.01496 - Zuevsky, 1 Sep 2026) in Section 5, immediately following Proposition 5.1