Flag Variety

A flag variety is an algebraic variety whose points correspond to complete flags of subspaces in a vector space — nested sequences of subspaces, one of each dimension, from zero up to the full space. The flag variety of a semisimple Lie group is a homogeneous space G/B, where G is the group and B is a Borel subgroup, and it carries the structure of a projective algebraic variety. Its Schubert cells, indexed by elements of the Weyl group, give a cell decomposition that makes the flag variety a central object in intersection theory and algebraic combinatorics. The flag variety is not merely a geometric object; it is the stage on which the representation theory of semisimple Lie groups is performed. The Borel-Weil Theorem shows that every irreducible representation of a compact semisimple Lie group can be realized as the space of holomorphic sections of a line bundle over the flag variety, making the variety itself a universal representation machine. The same construction extends to Kac-Moody groups, suggesting that the flag variety is a universal pattern, not a classical artifact.