Transfer Matrix Eigenvalue Polynomials for Quantum Affine Algebras

Definitions and mathematical statement

Let $V=V_1\otimes\cdots\otimes V_m$ be a tensor product of finite-dimensional modules over a quantum affine algebra, and let $W\subseteq V$ be a common generalized eigenspace for the transfer matrices under consideration. Here a common generalized eigenspace means a subspace associated with fixed generalized eigenvalues of the operators in the intended transfer-matrix family. The source denotes by $Q_{i,k}(z)$ the Baxter polynomials appearing in formulas for the joint eigenvalues associated with $W$, with $z$ their polynomial variable.

Requested characterization. Characterize the families of Baxter polynomials $Q_{i,k}(z)$ that occur in these joint transfer-matrix eigenvalue formulas for $W$.

Unresolved formulation. This is the mathematical target stated in the supplied record, but it is not a fully specified classification problem. The record does not specify the quantum affine algebra, its parameters, the coefficient field, the permitted modules, the transfer-matrix family or its eigenvalue formulas, the ranges and meanings of $i$ and $k$, or the definition and normalization of $Q_{i,k}(z)$. Consequently, the polynomial families to be characterized cannot be defined precisely from the supplied evidence. No particular QQ-system, recurrence, degree restriction, root condition, or normalization is imposed here.

Short English statement

The problem asks which Baxter polynomials $Q_{i,k}(z)$ appear in transfer-matrix eigenvalue formulas on a common generalized eigenspace in a tensor product of finite-dimensional quantum affine algebra modules. The supplied record leaves the polynomial family and the relevant formulas undefined, so a precise characterization problem requires further source clarification. Its open status is a supplied curation label.

Sources

• Local UnsolvedMath registry, permanent ID $30003495$, source problem code catalogue entry, title Transfer Matrix Eigenvalue Polynomials for Quantum Affine Algebras, in the Oberwolfach Reports Open Problems collection. No proposer is identified; the supplied proposed year is 2017.
• Supplied report citation: Algebraic Groups (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/21, pp. 1281--1347. https://doi.org/10.4171/owr/2017/21. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The literature-source label for this same link is Oberwolfach Reports 2018.
• Supplied literature reference: Frenkel et al., q-Opers and QQ-Systems. https://arxiv.org/abs/2002.07344.
• Supplied literature reference: On Wronskians and QQ-Systems. https://arxiv.org/abs/2208.08018.
• The literature assessment dated 2026-08-21 is supplied dated curation, not independently verified here. It labels the problem open and reports that later papers construct many polynomial QQ-system solutions and Wronskian recurrences, while the extracted question does not precisely define the family $Q_{i,k}$, preventing assignment of a general answer. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly and that the clean statement restored definitions while preserving the target. These are supplied verification claims; this draft does not independently verify them or claim to have visited any source.