The Terwilliger algebras of doubly regular tournaments

The Terwilliger algebras of asymmetric association schemes of rank 3, whose nonidentity relations correspond to doubly regular tournaments, are shown to have thin irreducible modules, and to always be of dimension \(4k+9\) for some positive integer k. It is determined that asymmetric rank 3 association schemes of order up to 23 are determined up to combinatorial isomorphism by the list of their complex Terwilliger algebras at each vertex, but this is no longer true at order 27. To distinguish order 27 asymmetric rank 3 association schemes, it is shown using computer calculations that the list of rational Terwilliger algebras at each vertex will suffice.