In more generality, if a matrix acts on a group of matrices by conjugation, what is the determinant of this action (if such a notion exists)? Is it simply the determinant of the matrix being used to conjugate?
In particular, given a Lie group $G$, with $g \in G$, we have a map $$ Int_g:G \to G, x \mapsto gxg^{-1}. $$ Differentiating this action gives a map $$ Ad_g: \mathfrak{g} \to \mathfrak{g}. $$ Thus we have a map $$ Ad: G \to Aut(\mathfrak{g}). $$ If $G = GL(n,\mathbb{C})$, we have that $G$ acts by $$ g \cdot X = gXg^{-1}, $$ for $g \in G, X \in \mathfrak{g}=\mathfrak{gl}(n,\mathbb{C})$.
A few of the sources I have been using refer to the determinant of this action; how is this calculated? Thanks in advance!
Let $g\in G$ and $f:X\in M_n\rightarrow gXg^{-1}$. Then $f=g\otimes g^{-T}$ -if we stack the matrices row by row- cf. http://en.wikipedia.org/wiki/Kronecker_product If $spectrum(g)=(\lambda_i)_i$, then $spectrum(f)=\{\lambda_i/\lambda_j|i,j\}$. Finally $\det(f)=1$.