doi:10.1093/imrn/rnq104 Skew Littlewood–Richardson Rules from Hopf Algebras

We use Hopf algebras to prove a version of the Littlewood–Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood–Richardson rules for Schur P- and Q-functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. Assaf and McNamara [1] recently used combinatorics to give an elegant and surprising “skew Pieri rule, ” a formula for the product of a skew Schur function and a complete homogeneous symmetric function. Their paper also included a conjectural skew version of the Littlewood–Richardson rule and an appendix by one of us (Lam) that provided a simple algebraic proof of their skew Pieri rule. In this paper, we extend this algebraic point of view further, showing that the Assaf–McNamara skew Pieri rule and conjectured skew Littlewood–Richardson rule are special cases of a simple formula holding for any pair of dual Hopf algebras. We establish this Hopf algebraic formula and explore its implications in some well-known Hopf algebras in combinatorics. In particular, we deduce the conjectura