Double coset Schur rings and diagonal subgroups

We investigate the structure of Schur rings arising from group triples $(G, H, K)$, where K H ≤ G, building on foundational work by Brender and Travis. Central to our approach is the interplay between Schur rings and (strong) Gelfand pairs. We introduce certain double coset algebras, and illustrate our findings with explicit computations for the general linear group GL(2,Fq), and its Borel subgroup. Finally, we bridge our algebraic results with combinatorial design theory by demonstrating that the structure constants of these double coset algebras yield the exact intersection numbers for novel, coarser (in the sense of fewer and larger classes) association schemes.