Association schemes on 28 points as mergings of a half-homogeneous coherent configuration

We consider a rank 112 coherent configuration S = A P (2) on 28 points with 7 fibers of size 4. We describe S both axiomatically and as a model arising via the regular action of E₈ on the set of all 2-element subsets of an 8-element set. Moreover, we prove that our model is the unique structure, up to isomorphism, which satisfies the established axioms. A most important feature of S is that its group A Aut (S) of algebraic automorphisms contains as a non-normal subgroup of index 8 the subgroup induced by all color automorphisms of S. This leads to a new type of automorphism of S, which we call "proper algebraic". All homogeneous mergings of S are described by us with the aid of a computer. Here, special attention is paid to so-called "algebraic mergings", i.e., those which arise from suitable subgroups of A Aut (S). As a result we are able to give a unified explanation of various association schemes on 28 points, including those of pseudocyclic and quasithin type, plus some of pseudotriangular type. Moreover, we provide computer-free proofs that these schemes are in fact attainable via appropriate mergings of classes from S. Another interesting phenomenon is the existence of many "twins", i.e., pairs of non-isomorphic association schemes which are algebraically isomorphic inside S. Notable examples of twins are the triangular graph T (8) paired with one of the Chang graphs, and the Mathon pseudocyclic scheme paired with the pseudocyclic scheme of Hollmann. In all, we decribe four pairs of twins and one set of triplets in rather great detail.