Let $G$ be a connected linear algebraic group (a torus, for example) of finite type over a field $K$. I have several, I suppose rather base, questions concerning the theory of such groups.
Suppose we fix an embedding $\iota\colon G\hookrightarrow GL_n$. In $GL_n$ we can define eigenvalues of the elements of $G$. Let us choose $g\in G$ with eigenvalues $\lambda_1,...\lambda_n$ and we define $K_{g,\iota}$ as $K(\lambda_1,...\lambda_n)$. Does $K_{g,\iota}$ depend on $\iota$? (EDIT) It depends. But here's another conjecture. There is a "universal" embedding $u\colon G\hookrightarrow GL_n$ such that $\forall \iota\colon G\hookrightarrow GL_m$ $\forall g$ $K_{g,\iota}\subset K_{g,u}$. Perhaps this embedding is unique. Perhaps it doesn't exist but the composite of all $K_{g,\iota}$ for a fixed $g$ is finite over $K$.
Suppose again we have an embedding $\iota\colon G\hookrightarrow GL_n$. Then we have the determinant map $\Delta\colon GL_n\to \mathbb G_m$. Does the map $\Delta\circ\iota$ depend on the choice of $\iota$? (EDIT) It depends. Just take the tensor product of two representations. But again, perhaps there exists a universal embedding $u\hookrightarrow GL_n$ such that for all other embeddings $\iota$ $\Delta\circ\iota = (\Delta\circ u)^m$ for some $m$. Perhaps an embedding doesn't exist but still there is a homomorphism $\alpha\colon G\to \mathbb G_m$ such that $\Delta\circ\iota = \alpha^m$. (EDIT 2) As Viktor Petrov said it's not true for tori as the lattice of their one-dimensional representations can have a big rank.
Let $G\hookrightarrow GL_n\cong GL(V)\subset \textrm{End}(V)$ where $\textrm{End}(V)$ is a ring algebraic variety of endomorphisms of $V$. Let us call $R_\iota$ the minimal ring variety containing $G$. Does $R_\iota$ depend on the embedding?
