Generic Bases for Cluster Algebras

Definitions and mathematical statement

The supplied record concerns a cluster algebra denoted by $A(B,x)$, with an initial tuple written $x=(x_1,\ldots,x_m)$. For nonnegative integer vectors $a=(a_1,\ldots,a_m)$ and $\mathbf d=(d_1,\ldots,d_m)$, write \[ x^a=\prod_{i=1}^m x_i^{a_i},\qquad a\cdot\mathbf d=\sum_{i=1}^m a_i d_i. \] Thus $a\cdot\mathbf d=0$ means that $a_i d_i=0$ for every $i$. Here nonnegative integer indexing is the interpretation used for the displayed exponents and dimension vectors.

The source uses $J$, $\operatorname{Rep}_{\mathbf d}(J)$, $\operatorname{Irr}_{sr}$, and $C_Z$ without supplying their definitions. In particular, it does not specify the algebra $J$ or its relation to $B$, the ground field for the representation spaces, the condition denoted by the subscript $sr$, or the construction of the element $C_Z$ associated with an indexed component $Z$. These symbols retain their source-dependent meanings here.

Conjecture as supplied. For the intended cluster algebra $A(B,x)$ and its intended associated data, the set \[ G=\left\{x^a C_Z\;\middle|\; a,\mathbf d\in\mathbb N^m,\quad a\cdot\mathbf d=0,\quad Z\in\operatorname{Irr}_{sr}\left(\operatorname{Rep}_{\mathbf d}(J)\right) \right\} \] is a basis of $A(B,x)$.

The coefficient ring over which this is a basis is not specified. If that ring is denoted by $R$, the basis assertion means that every displayed element belongs to $A(B,x)$, and every $f\in A(B,x)$ has a unique expression \[ f=\sum_{g\in G}r_g g,\qquad r_g\in R, \] with only finitely many nonzero coefficients. This explains the intended algebraic basis property conditionally; it does not supply the missing choice of $R$.

The permitted matrices $B$, coefficient or frozen-variable conventions, and hypotheses on any potential are not supplied. Although the background calls the target a generic-potential formulation, it gives neither the potential nor the meaning of genericity. Consequently, the displayed conjecture is a faithful statement template whose full hypotheses and quantification require source clarification.

Short English statement

The conjecture asks whether the displayed monomials times the source-dependent elements $C_Z$, indexed subject to disjoint exponent and dimension-vector supports, form a basis of the intended cluster algebra. The algebraic setup, indexing condition, construction of $C_Z$, and coefficient ring remain unspecified. The supplied dated curation labels the problem partially solved and reports acyclic special cases; these claims have not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001713$, exact source problem code catalogue entry, title Generic Bases for Cluster Algebras, in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2011.
• Supplied report citation: Representation Theory of Quivers and Finite Dimensional Algebras (2011), Oberwolfach Reports, pp. 523--608, DOI 10.4171/owr/2011/10. https://doi.org/10.4171/owr/2011/10. The record attributes the problem to workshop open-problem discussions documented by this report; no exact problem page or speaker is supplied.
• Supplied literature source: Geiss--Leclerc--Schröer, https://arxiv.org/abs/1004.2781. The record supplies no title, theorem number, or precise correspondence between this reference and the displayed formulation.
• Supplied literature source: Oberwolfach Report 10/2011, https://doi.org/10.4171/owr/2011/10.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and states that the generic-basis claim is proved for important acyclic cases, including the coefficient-free acyclic case, while the extracted generic-potential formulation is not established in full generality. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent verification.