Littlewood-richardson Rules for Grassmannians

The classical Littlewood-Richardson rule [LR] describes the structure constants obtained when the cup product of two Schubert classes in the cohomology ring of a complex Grassmannian is written as a linear combination of Schubert classes. It also gives a rule for decomposing the tensor product of two irreducible polynomial representations of the general linear group into irreducibles, or equivalently, for expanding the product of two Schur S-functions in the basis of Schur S-functions. In this paper we give a short and self-contained argument which shows that this rule is a direct consequence of Pieri’s formula [P] for the product of a Schubert class with a special Schubert class. There is an analogous Littlewood-Richardson rule for the Grassmannians which parametrize maximal isotropic subspaces of C n , equipped with a symplectic or orthogonal form. The precise formulation of this rule is due to Stembridge [St], working in the context of Schur’s Q-functions [S]; the connection to geometry was shown by Hiller and Boe [HB] and Pragacz [Pr]. The argument here for the type A rule works equally well in these more difficult cases and gives a simple derivation of Stembridge’s rule from the Pieri formula of [HB]. Currently there are many proofs available for the classical Littlewood-Richardson rule, some of them quite short. The proof of Remmel and Shimozono [RS] is also based on the Pieri rule; see the recent survey of van Leeuwen [vL] for alternatives. In contrast, we know of only two prior approaches to Stembridge’s rule (described in [St, HH] and [Sh], respectively), both of which are rather involved. The argument presented here proceeds by defining an abelian group H with a basis of Schubert symbols, and a bilinear product on H with structure constants coming from the Littlewood-Richardson rule in each case. Since this rule is compatible with the Pieri products, it suffices to show that H is an associative algebra. The proof of associativity is based on Schützenberger slides in type A, and uses the more general slides for marked shifted tableaux due to Worley [W] and Sagan [Sa] in the other Lie types. In each case, we need only basic properties of these operations which are easily verified from the definitions. Our paper is self-contained, once the Pieri rules are granted. The work on this article was completed during a fruitful visit to the Mathematisches Forschungsinstitut Oberwolfach, as part of the Research in Pairs program. It is a pleasure to thank the Institut for its hospitality and stimulating atmosphere.