A skew Murnaghan--Nakayama rule for Hopf dual pairs

Abstract

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 $(\mathrm{NSym},\mathrm{QSym})$ and $({\Lambda }^{(k)},{\Lambda }_{(k)})$ 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.

118 pages. v2: Added precise combinatorial connections to $k$-Schur theory and made several minor corrections