Schubert calculus is the study of flag varieties, which are quotients of algebraic groups (usually complex semisimple, but sometimes over the real numbers or even finite fields) by parabolic subgroups.
Schubert calculus is the study of flag varieties, which are quotients of algebraic groups (usually complex semisimple, but sometimes over the real numbers or even finite fields) by parabolic subgroups. Initially the topic only covered intersection theory on the Grassmannian. This culminated in the Littlewood-Richardson rule conjectured in 1939, finally proved by Marcel and Schützenberger in 1970. In the next couple of decades Littlewood-Richardson rules were found for maximal orthogonal Grassmannians, then Pieri formulas for arbitrary orthogonal Grassmannians. In type A the only additional further Littlewood-Richardson rule so far found was for the two-step flag variety in 2014, with a conjecture for the 3-step flag variety open since the 1990s, though a Pieri formula was found for the complete flag variety in the 1990s by Sottile. The topic is still an active area of interest where more general flag varieties are studied, as well as other associated algebraic structures such as torus-equivariant cohomology, K-theory, and equivariant K-theory rings.
