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arXiv · 2009.03159

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

Giusy Monzillo·Alessandro Siciliano
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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.

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Giusy Monzillo, Alessandro Siciliano. 2020-09-07. On the isomorphism of certain primitive $Q$-polynomial not $P$-polynomial association schemes. https://arxiv.org/abs/2009.03159

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