A Murnaghan--Nakayama rule for dual stable Grothendieck functions

Abstract

We give a closed boundary formula for multiplication by power sums in the dual stable Grothendieck basis. The off-diagonal coefficients are signed binomial coefficients supported on corner-connected broken ribbons, which are skew diagrams connected through edges or vertices and containing no square of four boxes. For such a diagram, the smallest power-sum index with a nonzero coefficient is the number of boxes plus the number of edge-connected components minus one. All larger indices give nonzero coefficients when the boundary statistic is positive and zero coefficients when it vanishes. The diagonal coefficient is minus the number of removable corners of the indexing partition. The formula evaluates the stable specialization of Nguyen's Murnaghan--Nakayama rule for K-k-Schur functions. We give two independent proofs, one using Newton's identity and the dual Pieri rules and the other using determinantal formulas and Schur complements.