Character Tables of Fission Schemes and Fusion Schemes

For given commutative association schemes H = (X, {Ri}0⩽i⩽d and Y = (X, {Sj}0⩽j⩽e, if {Ri}0⩽i⩽d is a refinement of {Sj}0⩽j⩽e, then we say that X is a fission scheme of Y. It is shown that if each class Sj of Y is either equal to Ri for some i or obtained by fusing two Riʹs in a simple uniform pattern, then the character table of one is determined by the other. This fact is proved by orthogonality relation of the character tables. Then the character table of the group scheme coming from the action of Chevalley group G2 (q) on the set Ω± of hyperplanes of type O±6(q) in the seven-dimensional orthogonal geometry over GF(q) is calculated from the character table of its fusion scheme, which is obtained from the action of O7(q) on Ω±, respectively. Further application of the main result is discussed in various other examples.