intersection theory is called Schubert calculus. Note given a sequence the Schubert class is typically denoted as just . Also, the Schubert classes given by a single integer, , are called special classes. Using the Giambeli formula below, all of the Schubert classes can be generated from these special classes. Explanation In order to explain the definition, consider a generic -plane : it will have only a zero intersection with for , whereas for . For example, in , a -plane is the solution space of a system of five independent homogeneous linear equations. These equations will generically span when restricted to a subspace with , in which case the solution space (the intersection of with ) will consist only of the zero vector. However, once , then and will necessarily have nonzero intersection. For example, the expected dimension of intersection of and is , the intersection of and has expected dimension , and so on. The definition of a Schubert cycle states that the first value of with is generically smaller than the expected value by the parameter . The -planes given by these constraints then define special subvarieties of . Properties Inclusion There is a partial ordering on all -tuples where if for every . This gives the inclusion of Schubert cyclesshowing an increase of the indices corresponds to an even greater specialization of subvarieties. Codimension formula A Schubert cycle has codimensionwhich is stable under inclusions of Grassmannians. That is, the inclusiongiven by adding the extra basis element to each -plane, giving a -plane, has the propertyAlso, the inclusiongiven by inclusion of the -plane has the same pullback property. Intersection product The intersection product was first established using the Pieri and Giambelli formulas. Pieri formula In the special case , there is an explicit formula of the product of with an arbitrary Schubert class given byNote . This formula is called the Pieri formula and can be used to determine the intersection product of any two Schubert classes when combined with the Giambelli formula. For exampleand Giambelli formula Schubert classes with tuples of length two or more can be described as a determinantal equation using the classes of only one tuple. The Giambelli formula reads as the equationgiven by the determinant of a -matrix. For example,and Relation with Chern classes There is an easy description of the cohomology ring, or the Chow ring, of the Grassmannian using the