MathDB

Frenkel’s root-multiplicity bound conjecture

Let g\mathfrak{g} be the rank-three hyperbolic Kac–Moody Lie algebra of type A1++A_1^{++}, with invariant bilinear form (⋅,⋅)(\cdot,\cdot), and let p(n)p(n) denote the ordinary partition function. The conjecture asserts that for every root α\alpha of g\mathfrak{g}, dim⁡gα≤p ⁣(1−(α,α)2)\dim \mathfrak{g}_\alpha \le p\!\left(1-\frac{(\alpha,\alpha)}{2}\right).

References

Primary source

arXiv

Additional references

computer-assisted-proofkacmoody-algebrasrepresentation-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to prove Frenkel’s bound for the fundamental hyperbolic example, but independent checking has not yet appeared.

Frenkel’s conjecture asks whether root-space multiplicities satisfy a partition-function upper bound. Earlier literature treated the rank-33 hyperbolic case as open, while the newest claim concerns the algebra A1++A_1^{++}.

Known results

  • The conjecture is known in the dimension-2626 case.
  • Kac, Moody, and Wakimoto showed that it fails for E10E_{10} at level 22.
  • A 2015 paper proved an analogous bound for symmetric rank-22 hyperbolic algebras, but not for the rank-33 algebra F\mathfrak F.

September 2026 claimed proof

Gabriel B. Legros’s preprint A proof of Frenkel's bound for the hyperbolic Kac–Moody Lie Algebra A1++A_1^{++} claims the bound dim⁡gα≤p(1−(α,α)/2)\dim \mathfrak g_\alpha\le p(1-(\alpha,\alpha)/2) for every root, using analytic estimates, Weyl reduction, and finite computation. The finite verification has not been independently formalized or checked.

Current status (as of September 2026): The A1++A_1^{++} case is claimed solved by a preprint, but remains unverified; the broader rank-33 case F\mathfrak F is recorded in earlier sources as open.

Sources

Solutions 0

No solutions have been posted yet.