Higher-rank extension of Mano’s monodromy factorization

Extend the factorization of connection matrices into those of two degree-one linear q-difference systems, established in the q-Painlevé VI setting, to higher-rank linear q-difference systems.

Background

The paper discusses how Mano’s analysis of q-Painlevé VI identifies a factorization of the connection matrix into connection matrices associated with two degree-one q-difference systems. Ohyama, Ramis, and Sauloy interpret this factorization as a localization of monodromy at pairs of intermediate singularities, namely the zeros of det A(z).

The unresolved problem is to generalize this factorization and its monodromy-localization interpretation from the known low-rank setting to higher-rank systems. The paper indicates that the isomonodromy transformations developed here, together with explicit q-Stokes analysis for the degenerate leading coefficient case, may provide tools for such a generalization.

References

Ohyama, Ramis and Sauloy further interpret this factorization as localizing monodromy at pairs of intermediate singularities, namely the zeros of \det A(z), and regard its extension to higher-rank systems as a major open problem .

— Isomonodromic Deformations for Linear $q$-Difference Systems of Degree One  (2609.02952 - Ma, 2 Sep 2026) in Introduction, Section 1