A skew Murnaghan--Nakayama rule for Hopf dual pairs

We develop a uniform skew Murnaghan–Nakayama theory for graded Hopf dual pairs equipped with a nondegenerate Hopf pairing. Using the completed Cauchy element, its grouplike factorization, and the resulting partial contraction operators, we establish a general skew Cauchy identity together with an abstract skew Murnaghan–Nakayama rule. Specializing this framework recovers and extends the classical skew Murnaghan–Nakayama rule for symmetric functions, and yields new skew Murnaghan–Nakayama formulas in several settings, including the dual pairs $(\mathrm{N}\mathrm{S}\mathrm{y}\mathrm{m},\mathrm{Q}\mathrm{S}\mathrm{y}\mathrm{m})$ and $({\mathrm{\Lambda }}^{(k)},{\mathrm{\Lambda }}_{(k)})$ arising in $k$-Schur theory, as well as the type $C$ affine Grassmannian context. As applications, we obtain generating functions for irreducible characters of Ariki–Koike algebras, including their type $A$ and type $B$ specializations, as well as Hecke–Clifford algebras and $\mathfrak{q}$-rook monoid algebras. We also give ribbon-tableau expansions for skew $(q,t)$-Kostka polynomials and for the entries of the inverse transition matrix, thereby answering a question of Carbonara (1998). Finally, by specializing the auxiliary alphabet $Y$ to sums of powers of primitive roots of unity, we derive a skew plethystic Murnaghan–Nakayama formula together with a Schur expansion for skew modular Schur functions; as a further consequence, we confirm Walker’s conjecture (1994) by showing that if the transition from the modular Schur functions to the Schur basis is trivial in the row indexed by $\lambda$, then $\lambda$ must be a $k$-core.