The Loop Murnaghan-Nakayama Rule

TL;DR

This paper presents a combinatorial proof of a generalized Murnaghan-Nakayama rule for loop Schur functions and introduces shifted loop Schur functions with similar properties.

Contribution

It provides the first combinatorial proof of the loop Schur functions generalization and defines shifted versions with analogous relations.

Findings

Established a combinatorial proof for the generalized rule

Defined shifted loop Schur functions and proved their properties

Extended classical symmetric function identities to the loop setting

Abstract

We give a combinatorial proof of a natural generalization of the Murnaghan-Nakayama rule to loop Schur functions. We also define shifted loop Schur functions and prove that they satisfy a similar relation.