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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.

A Generalization of the Murnaghan–Nakayama Rule for K - k -Schur and k -Schur Functions