Littlewood–Richardson rules for Grassmannians

Abstract

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 Cn, 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 givesa 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 com- patible with the Pieri products, it suffices to show thatH is an associative algebra. The proof of associativity is based on Schutzenberger 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.