A new shifted Littlewood-Richardson rule and related developments

Abstract

As Littlewood-Richardson rules compute linear representation theory of symmetric groups and cohomology of ordinary Grassmannians, shifted Littlewood-Richardson rules compute analogous projective representation theory of symmetric groups and cohomology of orthogonal Grassmannians. The first shifted Littlewood-Richardson rule is due to Stembridge (1989). We give a new shifted Littlewood-Richardson rule that is provably more efficient in some cases and is more convenient for hand calculations. Our rule builds on ideas of Lascoux-Schutzenberger (1981), Haiman (1989), and Serrano (2010). Our rule stems from a deeper understanding of the shifted plactic monoid in the form of a new axiomatization. We show that it is the largest monoid satisfying a short list of natural axioms inspired again by work of Lascoux and Schutzenberger. In addition, we obtain the first algebraic proof of Serrano's shifted Littlewood-Richardson rule (2010) and a new proof of the Hiller-Boe shifted Pieri rule (1986). Lastly, we explore the question of constructing a jeu de taquin theory via a rectification algorithm that computes mixed insertion, and find an algorithm that serves that purpose as long as a fixed order of slides is followed. From an algebraic perspective, the search for such a rectification algorithm was formulated by Cho (2013). To be specific, Cho proposed an open problem asking for a satisfactory definition of plactic skew Schur P-functions. We solve that problem using the interaction between the Sagan-Worley jeu de taquin and shifted plactic classes.