Schubert polynomials for the classical groups

Abstract

The task of a theory of Schubert polynomials is to produce explicit representatives for Schubert classes in the cohomology ring of a flag variety, and to do so in a manner that is as natural as possible from a combinatorial point of view. To explain more fully, let us review a special case, the Schubert calculus for Grassmannians, where one asks for the number of linear spaces of given dimension satisfying certain geometric conditions. A typical problem is to find the number of lines meeting four given lines in general position in 3-space (answer below). For each of the four given lines, the set of lines meeting it is a Schubert variety in the Grassmannian and we want the number of intersection points of these four subvarieties. In the modem solution of this problem, the Schubert varieties induce canonical elements of the cohomology ring of the Grassmannian, called Schubert classes. The product of these Schubert classes is the class of a point times the number of intersection points, counted with appropriate multiplicities. This reformulation of the problem, though one of the great achievements of algebraic geometry, is only part of a solution. It remains to give a concrete model for the cohomology ring that makes explicit computation with Schubert classes possible. As it happens, the cohomology rings of Grassmannians can be identified with quotients of a polynomial ring so that Schubert classes correspond to Schur functions. Intersection numbers such as we are considering then turn out to be Littlewood-Richardson coefficients. For example, the answer to our four-lines 4 problem is the coefficient of the Schur function s(2 2) in the product s(1) X or 2. For an extended treatment and history of the subject, see [10], [11], [18]. The identification of Schur functions as Schubert polynomials for Grassmannians is a consequence of a more general and now highly developed theory of Schubert polynomials for the flag varieties of the special linear groups SL(n, C). The starting point for this more general theory is a construction of