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Quantum Pfaffians in quantum cluster algebras (conjecture)

Determine whether quantum Pfaffians admit realizations as cluster variables in quantum cluster algebras associated with quantum Grassmannians or as elements of canonical bases of U_q(g)-modules, and characterize the precise correspondence if it exists.

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Background

The paper reviews how quantum coordinate rings and Grassmannians can carry quantum cluster algebra structures, suggesting a potential combinatorial framework for expressing quantum invariants.

Identifying quantum Pfaffians with cluster variables or canonical basis elements would provide structural and computational insights, linking braided multilinear invariants to cluster mutations and positivity phenomena.

References

Given the deep connections between quantum groups and cluster algebras, it is conjectured that quantum Pfaffians may admit interpretations in terms of cluster variables or canonical bases of U_q(\mathfrak{g}) -modules.

— A Quantum Analogue of the Pfaffian-Determinant Identity, An Algebraic and Geometric Study in the q-Skew-Symmetric Case (2508.11634 - Safadi, 24 Jul 2025) in Section 7.2 (Open Problems and Conjectures)