I am studying the following example on representation of general Linear groups. Before that I mention a brief description of representation of general linear groups in general framework. Let consider the representation $(V, \rho)$ of $GL(C^{n})$ , the group of invertible $n\times n$ matrices, where $\rho : GL(C^{n}) \to GL(V)$ such that $\rho (A)=(\rho_{kl}(A))_{1 \leq k,l \leq N}$ and $N$ is the dimension of vector space $V$. The following representation sometimes is called $\textit{polynomial representation}$ as well. We say that $(V, \rho)$ is a polynomial representation of degee $d$ if the matrix entries $\rho_{kl} (A) = \rho_{kl} (a_{11}, a_{12}, \dots, a_{nn})$ are polynomial functions.
Example. Let $V$ be the space of homogeneous polynomials of degree $3$ in $x,y$, so $V=span \{ x^3,x^2y,xy^2, y^3 \} \cong C^4$ and $\rho$ be the group homomorphism,then we have \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ %\hdotsfor{5} \\ % x_{d1} & x_{d2} & \end{bmatrix} $\to$ \begin{bmatrix} a_{11}^3 & a_{11}^2a_{12} & a_{11}a_{12}^2 & a_{12}^{3} \\ 3a_{11}^2a_{21} & a_{11}^2a_{22}+2a_{11}a_{12}a_{21} & 2a_{11}a_{12}a_{22}+a_{12}a_{21}^2 & 3a_{12}^2a_{22} \\ 3a_{11}a_{21}^2 & 2a_{11}a_{21}a_{22}+a_{12}a_{21}^2 & a_{11}a_{22}^2+2a_{12}a_{21}a_{22} & 3a_{12}a_{22}^2 \\ a_{21}^3 & a_{21}^2a_{22} & a_{21}a_{22}^2 & a_{22}^3 \end{bmatrix} I need an explicit explanation on the entries of $4 \times 4$ matrix. I appreciate your answer.