On the isomorphism of certain primitive $Q$-polynomial not $P$-polynomial association schemes

TL;DR

This paper proves that certain primitive Q-polynomial association schemes, previously thought to be distinct, are actually isomorphic, linking geometric constructions with algebraic group actions and establishing isomorphisms of related strongly regular graphs.

Contribution

It demonstrates the isomorphism between two previously separate constructions of primitive Q-polynomial association schemes, unifying geometric and algebraic approaches.

Findings

The schemes constructed by Penttila and Williford are isomorphic to those by Hollmann and Xiang.

An isomorphism of the associated strongly regular graphs is established.

The result confirms the equivalence of different construction methods for these schemes.

Abstract

In 2011, Penttila and Williford constructed an infinite new family of primitive $Q$-polynomial 3-class association schemes, not arising from distance regular graphs, by exploring the geometry of the lines of the unitary polar space $H(3,q^2)$, $q$ even, with respect to a symplectic polar space $W(3,q)$ embedded in it. In a private communication to Penttila and Williford, H.~Tanaka pointed out that these schemes have the same parameters as the 3-class schemes found by Hollmann and Xiang in 2006 by considering the action of $\mathrm{PGL}(2,q^2)$, $q$ even, on a non-degenerate conic of $\mathrm{PG}(2,q^2)$ extended in $\mathrm{PG}(2,q^4)$. Therefore, the question arises whether the above association schemes are isomorphic. In this paper we provide the positive answer. As by product, we get an isomorphism of strongly regular graphs.