Chow ring
Schubert calculus can be constructed using the Chow ring of the Grassmannian where the generating cycles are represented by geometrically meaningful data. Denote formula_1 as the Grassmannian of formula_2-planes in a fixed formula_3-dimensional vector space formula_4, and formula_5 its Chow ring; note that sometimes the Grassmannian is denoted as formula_6 if the vector space isn't explicitly given. Associated to an arbitrary complete flag formula_7formula_8and a decreasing formula_2-tuple of integers formula_10 whereformula_11there are Schubert cycles (which are called Schubert cells when considering cellular homology instead of the Chow ring) formula_12 defined asformula_13Since the class formula_14 does not depend on the complete flag, the class can be written asformula_15which are called Schubert classes. It can be shown these classes generate the Chow ring, and the associated intersection theory is called Schubert calculus. Note given a sequence formula_16 the Schubert class formula_17 is typically denoted as just formula_18. Also, the Schubert classes given by a single integer, formula_19, are called special classes. Using the Giambeli formula below, all of the Schubert classes can be generated from these special classes.