Flag variety

Summary

The flag variety of a vector space $V$ parametrizes chains of subspaces $V_1 \subset V_2 \subset \cdots$ of prescribed dimensions. It is a smooth projective homogeneous space for $\mathrm{GL}(V)$; projective spaces and Grassmannians are special cases. In general, a flag variety is a quotient $G / P$ of a reductive group by a parabolic subgroup. Flag varieties are where representation theory, combinatorics and geometry meet, through Schubert varieties and the Borel–Weil theorem.