Quantum Grothendieck Rings Beyond Simply-Laced Types

Definitions and mathematical statement

The supplied record concerns Conjecture 4 in the cited Oberwolfach report. Its intended category $\mathcal C$ consists of finite-dimensional representations of a quantum affine algebra. The record does not specify the algebra, its deformation parameter, the admissible categories, or the hypotheses needed for the constructions below. Consequently, the following is a conditional formulation of the supplied target, requiring the source definitions before it becomes a fully specified conjecture.

For an exact category $\mathcal C$, its Grothendieck group $K(\mathcal C)$ is generated by isomorphism classes $[V]$, subject to $[V]=[U]+[W]$ for each short exact sequence $0\to U\to V\to W\to0$ in its specified exact structure. The required exact and tensor structures on the intended $\mathcal C$ are not supplied. Denote the source's parameter-dependent Grothendieck module by $M=K(\mathcal C)_t$, its coefficient ring by $R$, and its quantum Grothendieck ring by $Q=K_t(\mathcal C)$. These are distinct source-designated objects; the record does not define their coefficient ring, parameter conventions, or underlying modules. Let $*:M\times M\to M$ denote the bilinear operation that the source constructs from a monoidal Jantzen filtration. The filtration and the formula for this operation are absent from the record.

Conditional conjectural target. For every category covered by the intended source hypotheses, the operation $*$ is associative: \[ (x*y)*z=x*(y*z)\qquad(x,y,z\in M). \] Together with the intended anti-involution $\iota$, it makes $M$ an algebra with anti-involution. In particular, $\iota$ is additive and satisfies \[ \iota^2=\operatorname{id}_M, \qquad \iota(x*y)=\iota(y)*\iota(x) \qquad(x,y\in M). \] The record does not specify whether $\iota$ is $R$-linear or semilinear for an involution of $R$, nor whether unitality is part of the asserted algebra structure.

There is an isomorphism of algebras with anti-involution \[ \Phi:(K(\mathcal C)_t,*,\iota)\xrightarrow{\sim}(K_t(\mathcal C),\cdot,\iota_Q) \] which identifies the corresponding standard bases. Thus, under the source's intended coefficient conventions, \[ \Phi(x*y)=\Phi(x)\Phi(y), \qquad \Phi\circ\iota=\iota_Q\circ\Phi. \] For each prescribed pair of corresponding standard bases, written schematically as $(b_\lambda)_{\lambda\in\Lambda}$ in $M$ and $(s_\lambda)_{\lambda\in\Lambda}$ in $Q$, the required identification is $\Phi(b_\lambda)=s_\lambda$ for every $\lambda\in\Lambda$. The bases, their indexing, their correspondence, and $\iota_Q$ are not specified in the supplied record; this notation supplies no additional choices.

The supplied dated curation reports the conjecture proved for simply-laced quantum loop algebras and additional special categories, while reporting associativity and the claimed isomorphism conjectural in the general non-simply-laced setting. These are supplied status claims, not independently verified mathematical conclusions.

Short English statement

The conjecture asks whether the product obtained from a monoidal Jantzen filtration is associative and yields the quantum Grothendieck ring, respecting anti-involutions and the corresponding standard bases. The supplied record omits the precise category hypotheses and construction conventions, so this is a conditional statement requiring source consultation and human mathematical review. Its partially solved status is supplied dated curation.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004893$, source problem code catalogue entry, title Quantum Grothendieck Rings Beyond Simply-Laced Types, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2021; no individual proposer is supplied.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2022), Oberwolfach Reports, pp. 2827--2891, DOI 10.4171/owr/2021/52. https://doi.org/10.4171/owr/2021/52. The original-statement field refers to Conjecture 4 but does not reproduce its mathematical content.
• Supplied literature reference: Fujita--Hernandez, Advances in Mathematics. https://doi.org/10.1016/j.aim.2026.110963.
• Supplied literature reference: Fujita--Hernandez, Monoidal Jantzen filtrations. https://arxiv.org/abs/2402.13544.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, reports proofs for simply-laced quantum loop algebras and additional special categories, and reports the general non-simply-laced associativity and isomorphism claims as conjectural. This assessment is supplied dated curation, not independently verified; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and states that the official report was checked directly and the clean statement restored definitions while removing extraction artifacts. This is supplied verification metadata; no independent source visit or verification is asserted here.