I am following the book Symmetry, Representations and Invariants and I have a confusion with the definition of a regular function on $\mathbf{GL}(n,\mathbb{C})$:
For the group $\mathbf{GL}(n,\mathbb{C})$, the algebra of regular functions is defined as $$\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]:= \mathbb{C}[x_{11},x_{12},\dots,x_{nn},\text{det}^{-1}].$$ This is a commutative algebra over $\mathbb{C}$ generated by the matrix entry functions $\{x_{ij}\}$ and the function $\text{det}^{-1}$, with the relation $\text{det}(x)\cdot \text{det}^{-1}(x)=1$ (where $\text{det}(x)$ is expressed as a polynomial in $\{x_{ij}\}$ as usual).
By this definition $\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]$ is a set of polynomials and $x_{11},x_{12},\dots,x_{nn}, \text{det}^{-1}$ are just symbols, i.e., a regular function is just a polynomial in $n^{2}+1$ variables; but with this I do not understand how $\mathcal{O}[\mathbf{GL}(n,\mathbb{C})]$ is generated by the functions $\{x_{ij}\}$ and the function $\text{det}^{-1}$. They mention that $x_{ij}$ is a matrix entry function, so imagine $x_{ij}\colon M_{n}(\mathbb{C})\to \mathbb{C}$ is given by $x_{ij}(\mathbf{A})=\mathbf{A}_{ij}$ and $\text{det}^{-1}\colon\mathbf{GL}(n,\mathbb{C})\to \mathbb{C}$ maps each invertible square matrix to one over its determinant.
I would appreciate your help.
The notation $\mathbf{C}[x_{11}, \dots, x_{nn}, \det^{-1}]$ is a slight notational abuse for $$\mathbf{C}[x_{11}, \dots, x_{nn}, t] / (t \Delta - 1)$$
where $\Delta \in \mathbf{C}[x_{11}, \dots, x_{nn}]$ is the polynomial giving the determinant of the $n \times n$ matrix with entries $x_{ij}$. So the ring is a quotient of an $(n^2 + 1)$-variable polynomial ring.