A note on canonical stable Grothendieck functions
Siddheswar Kundu
TL;DR
The paper addresses deriving Murnaghan-Nakayama type rules for stable Grothendieck polynomials and their canonical two-parameter generalizations. It adapts the classical MN proof framework to the K-theoretic setting, employing determinant identities and hook-valued tableaux to handle the combinatorics of $G^{(\beta)}_{\lambda}$ and $G^{(\alpha,\beta)}_{\lambda}$. The main contributions are (i) a MN-type rule for $G^{(\beta)}_{\lambda}$ with explicit coefficients, (ii) a MN-type rule for canonical $G^{(\alpha,\beta)}_{\lambda}$ via the operator $p^{\alpha}_k$, and (iii) a supporting determinant lemma $A_{\gamma}^{\beta}$ used in an inductive proof. When $\beta=0$, the results specialize to the classical MN rule for Schur functions, and the framework unifies K-theoretic stable Grothendieck functions with their canonical generalization. Overall, the work provides concrete, computable expansion rules for products with power sums in the realm of K-theoretic symmetric functions, linking combinatorial hook-valued tableaux with determinant-based representations.
Abstract
In this article, we offer a new way to prove the Murnaghan-Nakayama type rule for the stable Grothendieck polynomials, originally established by Nguyen-Hiep-Son-Thuy. Additionally, we establish a Murnaghan-Nakayama type rule for cannoical stable Grothendieck functions.