$G(k,n)$ is the complex grassmanian which is homeomorphic to the space of projections in $M\_{n}(\mathbb{C})$ with trace $k$. So we can Identify $G(k,n)$ with $$\{A\in M\_{n}(\mathbb{C})\mid A=A^{\*}=A^{2},\;\;trac(A)=k\}$$


With this matrix interpretation, we define two maps from $G(k,n)$ to $G(2k,2n)$. Our quesion is that :



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> Are these maps homotpic maps?:
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The maps are $f(X)=X\otimes I\_{2}$ and $g(X)=I\_{2}\otimes X$



Homotopy type of certain maps on complex grassmanian
Let $U$ be the matrix which sends $e\_i$ to $e\_{2i}$ and $e\_{i+n}$ to $e\_{2i+1}$ for $0\leq i<n$. Then $f(X)=Ug(X)U^{-1}$. You can join $U$ to $I$ by a path in $U(n)$, and this gives a homotopy between $f$ and $g$.


