Zeros and Poles of Quantum-Group R-Matrices

Definitions and mathematical statement

The supplied record concerns finite-dimensional modules $V,V'$ over a quantum group denoted by $U_q(\mathfrak g)$, and meromorphic intertwining operators written schematically as \[ R_{V,V'}(z)\colon V(z)\otimes V'\longrightarrow V'\otimes V(z). \] Here an intertwiner means a linear map commuting with the specified quantum-group action. A tensor-product module is reducible if it contains a nonzero proper submodule. The notation $V(z)$ denotes the source's parameter-dependent module; its construction is not supplied.

Source-level request. For finite-dimensional $U_q(\mathfrak g)$-modules $V,V'$, determine the zeros and poles of these meromorphic $R$-matrix intertwiners and describe their relation to reducibility of $V(z)\otimes V'$.

Unresolved formulation. This is a schematic research request, not a complete mathematical problem. The record does not specify $\mathfrak g$, the version of $U_q(\mathfrak g)$, the allowed values of $q$, the module category, the parameter space for $z$, the construction of $V(z)$, or the normalization and meromorphic realization of the $R$-matrix. It also does not define a zero of an operator-valued meromorphic family: vanishing of the entire operator and failure of invertibility are different conditions. Scalar normalization can change zeros and poles. No equivalence or other precise relationship between singularities and reducibility is stated. Consequently, the supplied evidence does not determine parameter ranges, exceptional values, or a quantified criterion relating these phenomena.

Short English statement

The source asks how zeros and poles of parameter-dependent quantum-group $R$-matrices relate to reducibility of tensor products of finite-dimensional modules. Essential hypotheses, normalization conventions, and the intended relationship are missing, so this remains a statement-only draft of an incomplete research question requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30001447$, source problem code catalogue entry, title "Zeros and Poles of Quantum-Group R-Matrices." The supplied record assigns the year 2010 and gives no individual proposer.
• Supplied report citation: Combinatorial Representation Theory (2010), Oberwolfach Reports, pp. 799--882, DOI 10.4171/owr/2010/15. Source URL and supplied literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2010/15. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions; no pinpoint passage or individual attribution is supplied.
• Supplied literature assessment, checked 2026-08-21: the status is labelled "open," but the assessment says the extracted passage provides motivation concerning zeros and poles of quantum-group $R$-matrices and tensor-product irreducibility rather than a complete question or theorem statement. This 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 reports that its mathematical content was checked against the official report, with notation, context, and question form normalized without changing the source-level claim. This verification claim is supplied metadata, not an independent verification performed for this draft.