Combinatorial refinement of major index for dihedral linear branching coefficients

Develop a statistic on the set of standard Young tableaux SYT(λ) of a partition λ ⊢ n that refines the major index within the congruence classes {T ∈ SYT(λ) | maj(T) ≡ 0 mod n} (and, when n is even, also within {T ∈ SYT(λ) | maj(T) ≡ n/2 mod n}) such that the resulting refined counts recover, respectively, the dihedral branching multiplicities d(λ,ξ)=⟨Res^{S_n}_{D_n} χ_λ, ξ⟩ for each linear character ξ of the dihedral group D_n (i.e., the characters mapping the rotation r to ±1 and the reflection s to ±1).

Background

Through known results linking cyclic branching coefficients to the major index distribution on standard Young tableaux, the authors observe that sums of dihedral linear branching coefficients coincide with particular major index congruence classes (e.g., a_λ0 and, for even n, a_λ{n/2}).

They ask for a refined combinatorial statistic that splits these sums to match each individual dihedral linear character’s multiplicity upon restricting χ_λ from S_n to D_n, thus seeking a tableau-theoretic model for the dihedral linear branching coefficients analogous to the cyclic case.

References

As we have seen that d_{\mathbb{1}{\mathbb{1}}(\lambda) + d_{\mathbb{1}{-\mathbb{1}}(\lambda) = a_\lambda0, d_{-\mathbb{1}{\mathbb{1}}(\lambda) + d_{-\mathbb{1}{-\mathbb{1}}(\lambda) = a_{\lambda}{n/2} (when n is even) and d_{\psi}j(\lambda) = a_\lambdaj = a_\lambda{n-j} for 1\leq j \leq \lfloor (n-1)/2 \rfloor, it is natural to ask the following question. Is there a combinatorial interpretation for the dihedral branching coefficients of S_n to the linear characters of D_n? More precisely, is there a statistic \mathrm{stat} on \SYT(\lambda) which refines the major index, i.e., on the set {T\in \SYT(\lambda) \mid \maj(T)\equiv 0 \mod n} (and similarly on the set {T\in \SYT(\lambda) \mid \maj(T)\equiv n/2 \mod n} when n is even) such that this statistic recovers the dihedral branching coefficients for the linear characters of D_n?

— On the Positivity of Dihedral Branching Coefficients of the Symmetric and Alternating Groups (2512.14381 - S, 16 Dec 2025) in Introduction, Question