Crystals and Schur \(P\)-positive expansions (Q5915698)

Summary: We give a new characterization of Littlewood-Richardson-Stembridge tableaux for Schur \(P\)-functions by using the theory of \(\mathfrak{q}(n)\)-crystals. We also give alternate proofs of the Schur \(P\)-expansion of a skew Schur function due to \textit{F. Ardila} and \textit{L. G. Serrano} [J. Algebr. Comb. 36, No. 3, 409--423 (2012; Zbl 1253.05137)], and the Schur expansion of a Schur \(P\)-function due to Stembridge using the associated crystal structures. Finally we introduce the notion of semistandard decomposition tableaux of a shifted skew shape and discuss its crystal structure.