# Why are these graphs coming from 9-dimensional alternating trilinear forms so symmetric?

Let $$\phi(x,y,z)$$ be an alternating trilinear form on a space $$V$$ over a field $$K$$.

Let $$u \in \mathbb{P}(V)$$ be a projective point over $$V$$, then we say that the rank of $$u$$ is equal to the rank of the alternating bilinear form $$\phi(u,.,.)$$.

To the form $$\phi$$ we can associate a graph $$G_\phi$$, whose the vertices are the projective points in $$\mathbb{P}(V)$$, and two vertices $$u,v$$ share an edge if the linear form $$\phi(u,v,.)$$ is identically zero.

For Section 7 of my work, I computed the restriction of these graphs to points of rank 4 for uniformly random $$9$$-dimensional alternating trilinear forms over finite fields of prime order $$p$$, and I stumbled on the following observation, which I cannot explain:

It seems that with high probability (the probability seems to go to 1 as $$p$$ grows), the graphs have Dihedral symmetry $$D_{N}$$, where $$N$$ is the number of points of rank 4. See, e.g., pictures below over GF(5).

Where does this come form? A guess would be that the automorphisms of the graph come from automorphisms of the trilinear form, but this is not the case. Usually, random forms $$\phi$$ have only a few (e.g. 2,3,4, or 6) automorphisms, but the graphs have much more automorphisms.

In fact, the automorphism group of the graph is not (always) a subgroup of $$GL(n,K)$$, because the rotation of the second picture below has order 29, but 29 does not divide $$|GL(9,5)| = 2^{25}×3^5×5^{36}×7×11×13^2×19×31^3×71×313×829×19531$$.

It is not so difficult to prove that the average (for random $$\phi$$) number of nodes in the graph is $$q^2 + O(q)$$, and the average number of edges is $$q^3/2 + O(q^2)$$ (Theorem 1 in my work). But the structure of the graph is still very mysterious to me. The pictures say that there is a free and transitive group action of the cyclic group of order $$N$$ on the $$N$$ points of rank $$4$$, but I have no idea what this group is and how it acts.

• In case the link you supplied rots sometime in the next decade, your paper at the link is titled Graph-theorethic Algorithms for the Alternating Trilinear Form Equivalence problem Nov 8, 2022 at 13:43
• What is so special about rank 4 and dimension 9? Do you have similarly unexpected symmetries for other ranks/dimensions? Nov 9, 2022 at 16:56
• @MarcoGolla For $n \leq 8$ there is a huge number of automorphisms, just because $|GL(n,q)|\approx q^{n^2} > |ATF(n,q)| = q^{\binom{n}{3}}$, so these behave differently. For $n \geq 10$ I don't see the symmetries (although I didn't do a lot of experiments, because the graphs become very large and hard to compute). For $n=9$ I focus on rank 4, because that is whp the smallest rank that appears. Nov 9, 2022 at 19:17
• So it appears that rank 4 and dimension 9 is special somehow. Nov 9, 2022 at 19:25

The rotations of the graphs I plotted are translations $$X \mapsto X+Y$$, and the reflections are just inversion $$X \mapsto -X$$. You get one inversion per choice of identity element.