A Ratio of Alternants Formula for Loop Schur Functions

TL;DR

This paper introduces a new ratio formula for loop Schur functions, extending classical symmetric function theory, and provides a novel proof of the loop Murnaghan--Nakayama rule.

Contribution

It establishes a ratio of loop alternants formula for loop Schur functions, enhancing the theoretical framework of loop symmetric functions.

Findings

Loop Schur functions expressed as a ratio of loop alternants.

New proof of the loop Murnaghan--Nakayama rule.

Extension of classical Schur function properties to the loop setting.

Abstract

Lam and Pylyavskyy introduced loop symmetric functions as a generalization of symmetric functions. They defined loop Schur functions as generating functions over semistandard tableaux with respect to a `colored weight,' and they proved a Jacobi--Trudi-style determinantal formula for these generating functions. We prove that loop Schur functions can be expressed as a ratio of `loop alternants,' extending the analogy with Schur functions. As an application, we give a new proof of the loop version of the Murnaghan--Nakayama rule.