Multiplicity-free geometric degeneration for the coproduct formula

Determine whether there exists a multiplicity-free geometric degeneration that implies the coproduct expansion for the classes represented by translated noncrossing-alternating-forest operators.

Background

The coproduct theorem gives an expansion of the coproduct of an operator of the form σ∂F\sigma\partial_F, where FF is a noncrossing alternating forest, as a sum of tensor products of similar operators. Every coefficient in this algebraic expansion is equal to 1.

The authors observe that this coefficient pattern suggests an underlying geometric explanation: a degeneration with no multiplicities whose components correspond to the terms in the coproduct. The existence of such a geometric degeneration is not established and is explicitly left open.

References

The fact that the coefficients are all $1$ in the expansion of the coproduct suggests there may be a multiplicity-free geometric degeneration that implies this result -- we leave this as an open question.

— Long range divided differences, clusters, and Graham-positivity  (2609.10735 - Spink et al., 9 Sep 2026) in Section 6, immediately following Theorem 6.5 (the coproduct theorem)