As the title says. Which finite projective planes admit a symmetric incidence matrix? I am not an expert in the field at all, but I consulted with one. He claimed that $PG(2, \mathbb F_q)$ can always have a symmetric matrix, but he was unsure about the non-Desarguesian planes.
I am mostly asking for reference (both for the Desarguesian and non-Desaruesian case), but any insight or (quick) proof is welcome.
The key word here is "polarity". A polarity of a projective plane with point set $P$ and line set $L$ is a map $\pi$ from $P \cup L$ to itself mapping points to lines and lines to points, such that $\pi^2 = \operatorname{id}$ and such that a point $p$ is incident with a line $L$ if and only if the line $p^\pi$ is incident with the point $L^\pi$. The existence of a polarity $\pi$ is equivalent with the existence of a symmetric incidence matrix.
Not all projective planes admit polarities. One such non-example is given by the Hall planes (see, e.g., https://en.wikipedia.org/wiki/Hall_plane), because these projective planes are not self-dual.
For a reference, I would try looking in Hughes and Piper's 1973 book on Projective Planes, but I haven't checked this myself.
