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We determine explicitly the space of invariant Hermitian and Kahler metrics on the flag manifold. In particular, we show that a Killing metric is not Kahler. The Chern forms are also computed in terms of the Maurer-Cartan form, and this calculation is used to prove that the flag manifold is projective algebraic. An explicit projective embedding of the flag manifold is also given.doi:10.1017/s0004972700016300 fatcat:wa56yd5h4zhevm33c7cnhxl5y4