I am reading a preprint of a book of Landsberg (http://www.math.tamu.edu/~jml/simonsclass.pdf, section 6.1.6) about geometric complexity theory. But no need to understand what this theory is to answer my question! It's just for reference.
Let $S^n(\mathbb{C}^{n^2})$ the space of homogeneous polynomials of degree $n$ in $n^2$ variables. If one consider $X_n = [x_{ij}]_{1 \leq i,j \leq n}$ as an abstract matrix of $n^2$ variables, then $\det_n$, the determinant of this matrix, is a polynomial of $S^n(\mathbb{C}^{n^2})$. I consider this polynomial because it is the one in the text, but I think my question holds for any polynomial of $\mathbb{P}S^n(\mathbb{C}^{n^2})$.
Landsberg talk about the action of $\text{End}(\mathbb{C}^{n^2})$ on a polynomial in $S^n(\mathbb{C}^{n^2})$, which is the action on each variable of the polynomial. Because this theory uses algebraic geometry (and for other reasons), it is better to consider the projective space $\mathbb{P}S^n(\mathbb{C}^{n^2})$ and the point $[\det_n]$.
But after this projectivization, he talks about the orbit $\text{End}(\mathbb{C}^{n^2}) \cdot [\det_n]$. How is this well-defined? My first thought was, for $X \in \text{End}(\mathbb{C}^{n^2})$, that $X \cdot [\det_n] = [X \cdot \det_n]$. But if we consider the null matrix $0_{n^2}$, then $0_{n^2} \cdot [\det_n] = [0]$, which is not defined as a point of $\mathbb{P}S^n(\mathbb{C}^{n^2})$.
What it is that I don't understand? Do I have to take a quotient of $\text{End}(\mathbb{C}^{n^2})$, in the same way $\text{PGL}(V)$ is a quotient of $\text{GL}(V)$? I don't think so, because Landsberg also consider the orbit $\text{GL}(\mathbb{C}^{n^2}) \cdot [\det_n]$ without talking about the projective general group. Can we define $[0]$? I'm not that good in projective spaces, but I don't think one can. And I don't see other possibilities!
Thanks in advance!
I also don't know what this means, for exactly the same reason as you. My guess is that you should take all the points of the form $[X \cdot \text{det}_n]$ where this makes sense as a point in projective space, that is, where $X \cdot \text{det}_n \neq 0$, and you just disregard the zeroes.