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Consider the functions $\rho_1:M_1(\mathbb C)\to M_2(\mathbb R)$ where $\rho_1(a+bi)=\begin{pmatrix} a&b\\ -b&a \end{pmatrix}$ and $\rho_2:M_2(\mathbb C)\to M_4(\mathbb R)$ where $\rho_2\begin{pmatrix} a+bi&c+di\\ e+fi&h+ji \end{pmatrix}=\begin{pmatrix} a&b&c&d \\ -b&a&-d&c \\ e&f&h&j \\ -f&e&-j&h \end{pmatrix}$ This can be generalized to functions $\rho_n:M_n(\mathbb C)\to M_{2n}(\mathbb R)$.

If $A\in GL_n(\mathbb C)$ then $\rho_n(A)\in GL_{2n}(\mathbb R)$ and we know $\det(A)\neq 0$ and $\det(\rho_n(A))\neq 0$.

In that situation is there anything more precise that can be stated about $\det(A)$ and $\det(\rho_n(A))$ beyond both of them being nonzero?

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    No. The right conjecture is $\det(\rho_n A) = |\det(A)|^2$.2011-12-29

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