Categorical commutativity of the genus-two functors

Prove that the derived endofunctors \(\mathcal{F}_{ij}\) categorifying the genus-two Macdonald operators commute up to a natural isomorphism, namely establish \(\mathcal{F}_{ij}\circ\mathcal{F}_{ik}\simeq\mathcal{F}_{ik}\circ\mathcal{F}_{ij}\) for all distinct indices \(i,j,k\).

Background

The paper constructs derived endofunctors Fij\mathcal{F}_{ij} on a derived category of graded integrable representations and proves that their Grothendieck-group classes are the genus-two Macdonald operators. Since the corresponding operators commute, their decategorified classes commute automatically; the unresolved problem is to lift this relation from equality of Grothendieck-group classes to an isomorphism of functors.

References

It is natural to ask whether the relations satisfied by these operators can be lifted to isomorphisms of functors. The commutativity of the operators \hat O_{A_{ij}} is proved in , so that the classes [\mathcal{F}_{ij}] commute. We expect this to hold on the categorical level.

— Categorification of the genus two DAHA  (2608.13278 - Arthamonov et al., 13 Aug 2026) in Section 1, Introduction, immediately following Theorem B