Closed-form coefficients for the product species Kα

Determine whether subgroup-lattice Möbius inversion yields a closed-form formula for the coefficients j^λ_{α,β} in the product K_α × K_β = ∑_λ j^λ_{α,β}K_λ, where K_α = X^n/G_α and G_α = ⟨σ_{a_1}⟩ × ··· × ⟨σ_{a_k}⟩ is a product of cyclic groups.

Background

The species C_α considered in the paper use a single cyclic group ⟨σα⟩ acting diagonally across all rows of an infinite periodic pattern. This cyclic structure has a unique subgroup of each order, allowing the coefficients bλ{α,β} to be obtained by Möbius inversion over the divisor lattice of a single integer.

The companion species K_α instead uses the direct product G_α = ⟨σ{a_1}⟩ × ··* × ⟨σ{a_k}⟩, corresponding to independent shifts in the rows. Although G_α × G_β remains abelian and stabilizers remain orbit-invariant, the relevant groups can have several subgroups of the same order. Consequently, the one-dimensional divisor-lattice argument no longer applies, and the problem asks whether Möbius inversion over the full subgroup lattice can still produce a closed formula for the structure coefficients jλ_{α,β}.

References

For K_α × K_β, the relevant group is G_α × G_β, where G_α = ⟨σ{a_1}⟩ × * * * × ⟨σ{a_k}⟩ is a product of cyclic groups but is not itself cyclic once two parts of α coincide. The group G_α × G_β is still abelian, so Lemma~\ref{lem:orbit_invariant} continues to hold and stabilizers remain orbit-invariant. However, Lemma~\ref{lem:unique_subgroup} fails: G_α may have several subgroups of the same order, so the M"obius inversion of Theorem~\ref{thm:mobius} must be carried out over the full subgroup lattice of G_α (a product of divisor lattices) rather than over the divisors of a single integer. Determine whether this subgroup-lattice M"obius inversion still yields a closed-form formula for the coefficients j\lambda_{\alpha,\beta} in K_\alpha \times K_\beta = \sum_\lambda j\lambda_{\alpha,\beta}K_\lambda.

— The Molecular Species $\mathbf{C}_α$: Geometric Realization and a Closed Formula for Kronecker Coefficients  (2609.05010 - Baolahy et al., 4 Sep 2026) in Open Problem 1, Section 4 (Open problems), labeled prob:K