Compute Frobenius homomorphisms on higher-rank web basis elements

Compute the Frobenius homomorphism on web basis elements for higher-rank quantum groups, particularly for ranks de3, to clarify its expected connections with canonical bases, positivity properties, and categorified structures.

Background

The paper studies the PGL_2 skein theory at roots of unity and computes the Frobenius homomorphism explicitly for web bases on the 3-punctured sphere and the closed torus. It places these results in the broader context of higher-rank skein algebras, where determining the Frobenius map on webs is expected to illuminate canonical-basis phenomena and related categorified structures.

The unresolved problem concerns quantum groups of rank higher than two, in contrast to the explicit PGL_2 computations carried out in the paper. The statement is presented as an important open problem rather than as a question resolved by the paper.