Polynomial representation of $GL(V)$: definition and examples. I was reading Etingof's section on polynomial representation of $GL(V)$ and I immediately got stucked in the definition.

We say that a finite dimensional representation $ Y $ of $ GL(V) $ is polynomial (
or rational, or algebraic) if its matrix elements are polynomial functions of the
entries of $g,g^{-1}, \ g \in GL(V) \ \ $(i.e., belong to $k[g_{ij}][1/det g])$

What I understood is ( let's say for example that $ V=k^n $ and so $ GL(V) = GL_n(k):=G$ with $k$ algebraically close field):
If $Y$ is a representation of $G$, $G$ acts linearly on $Y$, so every element $g \in G$ defines an endomorfism of $Y$, thus to every element $ g \in G$ is associated a matrix $ M_g \in GL(Y) $. If the entries of this matrix are polynomial functions in the entries of $g,g^{-1}$ ( i.e., they lie in $k[(g)_{ij},(g^{-1})_{ij}: 0 \leq i,j \leq n]$, we say that $ Y $ is polynomial.
Is this correct?
Furthermore, does the definition of polynomial representation for $GL_n(k)$ easily extend to polynomial representation of   $ \  \prod_{i=1}^m GL_{n_i}(k)$ ? How?
Just to be clear, I'm studying quiver representation, and I need this notions to solve Problem 5.2 in Etingof's notes, that seems impossible to me right now.
Thanks to everybody.
 A: You are correct. However, we can (as quoted) get along with the $g_{ij}$ and $\frac1{\det g}$ alone as the $(g^{-1})_{ij}$ are themselves polynomials in the $g_{ij}$ and $\frac1{\det g}$.
Actually, I prefer to define $GL$ as an algebraic group, i.e., as a variety together with a multiplication and inverses map, by including $\frac1{\det g}$ as $(n^2+1)$st variable because that matches better with all variables of the product / of the inverse being polynomial expressions in the variables of the factors, i.e., without assigning a special role to the determinant (or to GL). For example, $GL_2$ is the variety defined by the single polynomial $X_{11}X_{22}Y-X_{12}X_{21}Y-1$ where $Y$ plays the role of $\frac1{\det\begin{pmatrix}X_{11}&X_{12}\\X_{21}&X_{22}\end{pmatrix}}$, and e.g. the inverses map is polynomial as follows: $(X_{11},X_{12},X_{21},X_{22},Y)\mapsto (X_{22}Y,-X_{12}Y,-X_{21}Y,X_{11}Y,X_{11}X_{22}-X_{12}X_{21})$.
With this in mind, the definition of polynomial representation of an algebraic group in terms of all associated matrices being polynomial in the group variables generalizes to arbitrary algebraic groups (in particular, to products of $GL$'s).
