A ratio of alternants formula for loop Schur functions

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–Trudistyle 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.

Keywords

loop symmetric function, Schur function, birational R-matrix, bialternant formula, Murnaghan–Nakayama rule

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The author was supported in part by NSF grants DMS-1464693 and DMS-0943832.

Received 17 May 2018

Published 14 January 2020