Quantum symmetry of $3$-transitive graphs

TL;DR

This paper determines the quantum automorphism groups of all classified 3-transitive graphs, revealing cases with and without quantum symmetry, and employs planar algebras to derive these results.

Contribution

It computes the quantum automorphism groups of all 3-transitive graphs, excluding certain orthogonal graphs, using planar algebra techniques.

Findings

No quantum symmetry for the McLaughlin graph.

No quantum symmetry for orthogonal graphs $ ext{O}^-(6,q)$ with $q=2,3$.

Quantum automorphism groups of affine polar graphs are monoidally equivalent to classical groups.

Abstract

We study the quantum automorphism group of $3$-transitive graphs in this article. Those are highly symmetric graphs that were classified by Cameron and Macpherson in 1985, and we compute the quantum automorphism group of all such graphs, excluding the orthogonal graphs $\mathrm{O}^-(6,q)$ for $q>3$. We show that there is no quantum symmetry for the McLaughlin graph and the orthogonal graphs $\mathrm{O}^-(6,q)$ with $q = 2, 3$, while that the quantum automorphism group of the affine polar graphs $\mathrm{VO}^{+}(2k,2)$ and $\mathrm{VO}^{-}(2k,2)$ are monoidally equivalent to $\mathrm{PO}(n)$ and $\mathrm{PSp}(n)$, respectively. We use planar algebras to obtain our results, where the $3$-transitivity of the graphs gives bounds on the dimensions of the $2$-- and $3$-box spaces of the associated planar algebras.