Braid-Group Extensions of Zamolodchikov Actions

Definitions and mathematical statement

Let $\Delta$ and $\Delta'$ be simply laced Dynkin diagrams. Write $h(\Gamma)$ for the Coxeter number of a Dynkin diagram $\Gamma$, and put \[ k=h(\Delta),\qquad n=k+h(\Delta'),\qquad d=\gcd(k,n). \] Thus $h(\Delta')=n-k$. Denote the cluster algebra in the supplied statement by $\mathcal A=\mathcal A(\Delta\square\Delta')$, and denote its Zamolodchikov transformation by $\rho$. Let $C_\rho=\langle\rho\rangle$ be the cyclic group of transformations generated by $\rho$.

Conjecture as supplied. For every such pair $(\Delta,\Delta')$, does the cyclic action of $C_\rho$ on $\mathcal A$ extend to an action of the extended braid group $\widetilde{\operatorname{Br}}_d$ on $\mathcal A$?

Unresolved specifications. The record does not define the product construction $\Delta\square\Delta'$, its required quiver orientations, the coefficient convention for $\mathcal A$, the transformation $\rho$, or the class of transformations allowed in the proposed action. Its original wording uses $\Delta\otimes\Delta'$ in place of $\Delta\square\Delta'$; their intended relationship requires confirmation. The record also supplies no presentation of $\widetilde{\operatorname{Br}}_d$, no conventions for small values of $d$, and no identification specifying how the cyclic action is to be recovered from that group.

For precision, if extension is intended through a specified homomorphism $\iota:C_\rho\to\widetilde{\operatorname{Br}}_d$, and $\operatorname{Aut}_{\mathrm{allowed}}(\mathcal A)$ denotes the intended group of allowed transformations, the extension condition would be the existence of a homomorphism \[ \Phi:\widetilde{\operatorname{Br}}_d\longrightarrow \operatorname{Aut}_{\mathrm{allowed}}(\mathcal A) \quad\text{such that}\quad \Phi\circ\iota=\alpha, \] where $\alpha$ is the given cyclic action. Neither $\iota$ nor the allowed transformation group is specified in the record. This conditional explanation does not supply the missing conventions or assert that an embedding of the cyclic group is intended.

Short English statement

The supplied conjecture asks whether, for every pair of simply laced Dynkin diagrams, the cyclic Zamolodchikov action on their associated cluster algebra extends to the indicated extended braid group, with index $d=\gcd(h(\Delta),h(\Delta)+h(\Delta')))$. The product, group, and extension conventions require clarification before this becomes a fully specified mathematical assertion.

Sources

• Local UnsolvedMath registry, permanent ID $30004363$, source problem code catalogue entry, title Braid-Group Extensions of Zamolodchikov Actions; supplied complete source record from the Oberwolfach Reports Open Problems collection. The project registry paths are data/unsolvedmath/problems.json and catalog/problems.json; the supplied evidence does not identify which contains this record.
• Supplied report citation: Representation Theory of Quivers and Finite Dimensional Algebras (2021), Oberwolfach Reports, pp. 143--230, DOI 10.4171/owr/2020/3. https://doi.org/10.4171/owr/2020/3. The record gives the proposed year as 2020 and names no individual proposer.
• Supplied literature reference: Fraser, Braid group symmetries of Grassmannian cluster algebras. https://doi.org/10.1007/s00029-020-0542-3.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the braid-group extension is verified for selected cluster-algebra families, while the conjecture for all products of Dynkin quivers has not been proved. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent checking.
• The supplied statement status is corrected_verified, with a supplied assertion that the clean statement was checked against the official Oberwolfach report and preserves its hypotheses, notation, and target. That verification assertion is retained as source metadata and has not been independently checked here.