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I have come across two different definitions of what I suspect is the same object. Both are called the complex Grassmannian:

1: $U(n)/U(k)\times U(n-k)$ 2: $SU(n)/(S(U(k)\times U(n-k))$

What is the difference between these two things? If they are the same, why is it true?

Finally, I would also like to know if there is more meaning to $S(U(k)\times U(n-k))$ than meets the eye. By this I mean the following: I tend to view this subgroup as elements of $SU(n)$ with four blocks: a $k$ by $k$ top left block which is made up of unitary group elements, an $n-k$ by $n-k$ block made of unitary group elements in the bottom right, and 0s in the remaining entry. There is one more additional restriction, which is that the whole matrix has determinant 1. Is there another interpretation?

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