Congruences of shifted Jack Littlewood-Richardson coefficients

Abstract

The shifted Jack Littlewood-Richardson coefficients $g^λ_{μν}(α)$, first studied by Alexandersson-Féray, are Laurent polynomials in the Jack parameter $α$ attached to triples of partitions, which generalize the classical Jack Littlewood-Richardson coefficients investigated by Stanley, et al. In a previous work of the author's, it was conjectured that the Littlewood-Richardson coefficients for two triples, in which one of the partitions differ by a single box move, are congruent modulo the $α$-hook length of the pivot box for that move. In this note we prove that conjecture. We also investigate the extension of that conjecture to shifted Macdonald functions, which remains open pending two properies of Lassalle's shift map in that case.