Journal of Combinatorics

Volume 11 (2020)

Number 2

A ratio of alternants formula for loop Schur functions

Pages: 359 – 376



Gabriel Frieden (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


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.


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

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

Received 17 May 2018

Published 14 January 2020