A four-class subscheme of the association of scheme coming from the action of \(\text{PGL} (2,4^f)\) (Q1348774)

Let \(\Omega\) denote the set of two-element subsets of the projective line \(\text{PG}(1,q)\), where \(q=2^e\). Then \(G=\text{PGL}(2,q)\) acts naturally on \(\Omega\times \Omega\) and the orbits of \(G\) acting on \(\Omega\times \Omega\) form the relations of the association scheme \(X(G,\Omega)\). Let \(H\) be the stabilizer of \(\{\langle (1,0)\rangle, \langle (0,1)\rangle\}\) in \(G\). Then \(X(G,\Omega)\) is the same as \(X(G,G/H)\), and this association scheme is symmetric. In this paper the order of the intersection of each conjugacy class of \(G\) and each double coset of \(H\) in \(G\), and the decomposition of the permutation character \(1_H^G\) into irreducible characters are determined. As a corollary, the character table of \(X(G,\Omega)\) is calculated. Further, it is proved that the conjectured subscheme of de Caen and van Dam of \(X(G,\Omega)\), \(q=4^f\) is indeed an association scheme (Theorem 4.1).