Let $ mathcal B $ to be the full flag variety of $ mathbb {C} ^ n $, and $ B_n $ the group of invertible upper triangular matrices.

Which brings up this question is the following statement:

Number of $ GL_n $ orbit $ mathcal B times mathcal B $ is finished.

This number is equal to the number of $ B_n $ orbit $ mathcal B $. Each flag $ F in mathcal B $ is stabilized by the conjugate of some $ T in B_n $ – since the dimension of $ mathcal B $ East $ n (n-1) / $ 2, there is $ n choose {2} $ Phone $ B_n $– orbits in $ mathcal B $.

Hence the following statement. Is this proof correct? Or is something missing in the plan above?