Class 3 association schemes whose symmetrizations have two classes

For a given commutative association scheme L, by fusing all the non-self-paired relations pairwise with their symmetric counterparts, we can obtain a new symmetric association scheme L̃. In this paper, we investigate when a symmetric association scheme L̃ of class 2 admits a symmetrizable commutative fission scheme L of class 3. By studying the feasible parameter sets obtained from the acceptible fission character tables of given symmetric schemes, we discover that there are infinitely many symmetric association schemes of class 2, each of which admits one or two symmetrizable commutative fission schemes of class 3. We then characterize and classify some such symmetric schemes and their symmetrizable commutative fission schemes. Also we prove that none of the Hamming schemes H(2, q) with q ≥ 3, and none of the Johnson schemes J(ν, 2) with ν ≥ 5 and ν ≢ 7(mod 8) have such commutative fission schemes of class 3. H(2, 2) has a symmetrizable commutative fission scheme which is isomorphic to the group scheme L(Z4), and J(4, 2) has one such fission scheme coming from the action of the alternating group A4 on the set of two-element subsets from a four-element set.