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

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.

22 pages, 6 pictures

References in corpus (2)

• The Murnaghan-Nakayama rule for k-Schur functions
• A Murnaghan-Nakayama rule for Grothendieck polynomials of Grassmannian type

Cited by in corpus (1)

• A quantum Murnaghan--Nakayama rule for the flag manifold