Let mathpzc K be a field of characteristic 0 and let varepsilon∈mathpzc K^× be not a root of unity. For every integer n≥ 2 and every integer r≥ 0, the natural algebra homomorphism xi_r: H_!vartriangle!(r)_mathpzc Klongrightarrow operatornameEnd_ S_!vartriangle!(n,r)_mathpzc K!(Omega_mathpzc K^otimes r)^op is an isomorphism.

Conjecture 3.8.8 of DDF and the DDF center conjecture

Let $\mathpzc K$ be a field of characteristic $0$ and let $\varepsilon\in\mathpzc K^{\times}$ be not a root of unity. For every integer $n\geq 2$ and every integer $r\geq 0$ , the natural algebra homomorphism $\xi_r:{\mathcal H}_{{\!\vartriangle\!}}(r)_{\mathpzc K}\longrightarrow\operatorname{End}_{{\mathcal S}_{{\!\vartriangle\!}}(n,r)_{\mathpzc K}}\!\left(\Omega_{\mathpzc K}^{\otimes r}\right)^{\mathrm{op}}$ is an isomorphism.

References

Primary source

arXiv

Additional references

Affine quantum Schur--Weyl duality — arXiv — Qiang Fu, Jun Hu

quantum-algebra representation-theory

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A September 2026 preprint claims to settle the conjectures, subject to explicit restrictions, but its proof has not been independently verified.

This entry concerns Conjecture $3.8.8$ of DDF and the DDF center conjecture, which form conjectural parts of affine quantum Schur–Weyl duality. The retrieved material does not identify the original proposers or date.

September 2026 claimed proof

Qiang Fu and Jun Hu’s preprint Affine quantum Schur--Weyl duality claims faithfulness of the natural affine Hecke action over general coefficient rings, and proves the centralizer isomorphism and related center conjecture in the stated field setting. It also reports Noetherianity results. The claim is unverified.

Current status (as of September 2026): The conjectures are claimed solved by Fu and Hu under stated restrictions, but independent verification remains outstanding.

Sources

No solutions have been posted yet.