I've defined a projective variety in Sage, and I would now like to act on that by matrices in $\operatorname{PSL}_n(\mathbb{F}_p)$. Unfortunately, I can't figure out how to create an element of $\operatorname{PSL}_n(\mathbb{F}_p)$ as a matrix, rather than as a permutation. My code for constructing the variety is below:
P.<a,b,c> = ProjectiveSpace(2, GF(11))
V = P3.subscheme([a^2 + b^2 - c^2])
I've figured out that I can construct $\operatorname{PSL}_3(\mathbb{F}_{11})$ as a permutation group with
G = PSL(3,11)
What I have, however, are particular matrices in $\operatorname{PSL}_3(\mathbb{F}_{11})$ that I would like to have act on the points in my variety V. Is there any way I can construct PSL as a matrix group, or convert my matrices to elements in G and have them act on the points of V?
The implementation of the finite projective linear groups comes from GAP, where they’re realized as permutations only. You can either circumvent the problem and work with the action of $\mathrm {SL}_3(\mathbb{F}_{11})$, or create your own $\mathrm{PSL}_n$ with matrices, just like it is done here.