Flag Varieties and the Bruhat Decomposition MATH 7895 Lecture

For our first example, consider V = Cn. Let {e1, . . . en} be the standard basis of Cn, and fix the full flag F• = 0 ⊂ 〈e1〉 ⊂ 〈e1, e2〉 ⊂ · · · ⊂ 〈e1, . . . , en−1〉 ⊂ V Note that any other flag can be obtained from this one by acting with an element of GLn(C). Specifically, any flag V• = 0 ⊂ V1 ⊂ · · · ⊂ Vn = V in Cn has the form V• = 0 ⊂ 〈ge1〉 ⊂ 〈ge1, ge2〉 ⊂ · · · ⊂ 〈ge1, . . . , gen〉 = V for some g ∈ GLn(C). From this we see that GLn(C) acts transitively on the set of full flags in Cn. Under this action, the stabilizer of our standard flag F• is the Borel subgroup of upper triangular matrices in GLn(C). We would like make sense of the statement GLn(C)/B = {flags in C},