The Murnaghan-Nakayama rule for k-Schur functions

TL;DR

This paper establishes a formula for expanding the product of a power sum symmetric function and a k-Schur function, using noncommutative algebraic tools, advancing the combinatorial understanding of symmetric functions.

Contribution

It proves the Murnaghan-Nakayama rule for k-Schur functions, providing an explicit expansion formula using noncommutative symmetric functions and the nilCoxeter algebra.

Findings

Derived an explicit expansion formula for k-Schur functions

Connected k-Schur functions with noncommutative algebraic structures

Extended classical symmetric function rules to the k-Schur setting

Abstract

We prove the Murgnaghan--Nakayama rule for $k$-Schur functions of Lapointe and Morse, that is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a $k$-Schur function in terms of $k$-Schur functions. This is proved using the noncommutative $k$-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene.