Giambelli, Pieri, and Tableau Formulas via Raising Operators

We give a direct proof of the equivalence between the Giambelli and Pieri type formulas for Hall-Littlewood functions using Young’s raising operators, parallel to joint work with Buch and Kresch for the Schubert classes on isotropic Grassmannians. We prove several closely related mirror identities enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert classes. The raising operator approach is applied to obtain tableau formulas for the Hall-Littlewood functions, the theta polynomials of [BKT2], and related Stanley symmetric functions. Finally, we introduce the notion of a skew element w of the hyperoctahedral group and identify the set of reduced words for w with the set of standard k-tableaux on a skew shape λ/μ. 0. Introduction The classical Schubert calculus is concerned with the algebraic structure of the cohomology ring of the Grassmannian G(m,N) ofm-dimensional subspaces of complex affine N -space. The cohomology has a free Z-basis of Schubert classes σλ, induced by the natural decomposition of G(m,N) into a disjoint union of Schubert cells. On the other hand, the ring is generated by the Chern classes of the universal quotient bundle Q over G(m,N), also known as the special Schubert classes. The theorems of Pieri [Pi] and Giambelli [G] are fundamental building blocks of the subject: the former is a rule for a product of a general Schubert class σλ with a special one, while the latter is a formula equating σλ with a polynomial in the special classes. When one expresses the Chern classes involved in terms of the Chern roots of Q, the Schubert classes are replaced by Schur S-polynomials, thus exhibiting a link between the Schubert calculus and the ring of symmetric functions. In a series of papers with Buch and Kresch [BKT1, BKT2], we obtained corresponding results for the Grassmannians parametrizing (non-maximal) isotropic subspaces of a vector space equipped with a nondegenerate symmetric or skewsymmetric bilinear form. Our solution of the Giambelli problem for isotropic Grassmannians uses the raising operators of Young [Y] in an essential way; the resulting formula interpolates naturally between a Jacobi-Trudi determinant and a Schur Pfaffian. A rather different context in which a Giambelli type formula appears that has this interpolation property is the theory of Hall-Littlewood symmetric functions [Li, M]; these objects also satisfy a Pieri rule [Mo]. The raising operator approach allows one to see directly that the Pieri and Giambelli results are formally equivalent to each other in all the above instances. This amounts to showing that the Giambelli polynomials satisfy the Pieri rule, for the Date: December 2, 2008. 2000 Mathematics Subject Classification. Primary 05E15; Secondary 14M15, 14N15, 05E05. The author was supported in part by NSF Grant DMS-0639033. 1