A Ratio of Alternants Formula for Loop Schur Functions

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.

arXiv comment12 pages. (v2) Added a denominator formula (Corollary 3.2) and an alternative ratio of alternants formula (Theorem 3.3); improved the exposition; changed notational conventions for $h_k^{(r)}$ and alternants