A generalization of the Murnaghan-Nakayama rule for $K$-$k$-Schur and $k$-Schur functions

TL;DR

This paper introduces a new family of symmetric functions that generalize $K$-$k$-Schur and $k$-Schur functions, providing a unified Murnaghan-Nakayama rule with explicit algorithms and recovering known results as special cases.

Contribution

The authors define a new symmetric function family $\\mathcal{F}_\lambda^{(k)}$ that generalizes existing $K$-$k$-Schur and $k$-Schur functions and establish a generalized Murnaghan-Nakayama rule for them.

Findings

Derived explicit Murnaghan-Nakayama rule for generalized functions.

Provided algorithms for computing coefficients in the rule.

Unified the $K$-$k$-Schur and $k$-Schur$ cases under a common framework.

Abstract

The $K$-$k$-Schur functions and $k$-Schur functions appeared in the study of $K$-theoretic and affine Schubert Calculus as polynomial representatives of Schubert classes. In this paper, we introduce a new family of symmetric functions $\mathcal{F}_\lambda^{(k)}$, that generalizes the constructions via the Pieri rule of $K$-$k$-Schur functions and $ k$-Schur functions. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule is described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases and connections promise to be detailed in the future.