Pieri-type Formulas for Maximal Isotropic Grassmannians via Triple Intersections

We give an elementary proof of the Pieri-type formula in the cohomology of a Grassmannian of maximal isotropic subspaces of an odd orthogonal or symplectic vector space. This proof proceeds by explicitly computing a triple intersection of Schubert varieties. The decisive step is an exact description of the intersection of two Schubert varieties, from which the multiplicities (which are powers of 2) in the Pieri-type formula are immediately obvious. Introduction The goal of this paper is to give an elementary geometric proof of Pieri-type formulas in the cohomology of Grassmannians of maximal isotropic subspaces of odd orthogonal or symplectic vector spaces. For this, we explicitly compute a triple intersection of Schubert varieties, where one is a special Schubert variety. Previously, Sertöz [16] had studied such triple intersections in orthogonal Grassmannians, but was unable to determine the intersection multiplicities and obtain a formula. These multiplicities are either 0 or powers of 2. Our proof explains them as the intersection multiplicity of a linear subspace (defining the special Schubert variety) with a collection of quadrics and linear subspaces (determined by the other two Schubert varieties). This is similar to triple intersection proofs of the classical Pieri formula (cf. [7][5, p. 203][4, §9.4]) where the multiplicities (0 or 1) count the number of points in the intersection of linear subspaces. A proof of the Pieri-type formula for classical flag varieties [17] was based upon those ideas. Similarly, the ideas here provide a basis for a proof of Pieri-type formulas in the cohomology of symplectic flag varieties [1]. These Pieri-type formulas are due to Hiller and Boe [6], whose proof used the Chevalley formula [2]. Another proof, using the Leibnitz formula for symplectic and orthogonal divided differences, was given by Pragacz and Ratajski [14]. These formulas also arise in the theory of projective representations of symmetric groups [15, 9] as product formulas for Schur P and Q-functions, and were first proven in this context by Morris [11]. The connection of Schur P and Q-functions to geometry was established by Pragacz [12] (see also [13]). In Section 1, we give the basic definitions and state the Pieri-type formulas in both the orthogonal and symplectic cases, and conclude with an outline of the proof in the Date: February 1, 2008. 1991 Mathematics Subject Classification. 14M15.