Determinant character (awaiting review)

To each mod-$\ell$ Galois representation $\rho:\Gal_K \to G(\F_{\ell})$, with values in the $\F_{\ell}$-points of an algebraic group $G$, is associated a determinant character $\det\rho:\Gal_K \to \F_{\ell}^*$, defined by composing $\rho$ with the determinant map $G(\F_{\ell}) \to \F_{\ell}^*$.

The determinant character is a $1$-dimensional mod-$\ell$ Galois representation. If $\rho$ has dimension $1$, the determinant character is the same as the original representation.