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Multiplicities corresponding to irreducible characters are defined for reality-based algebras. These algebras with a distinguished basis include fusion rings, C-algebras, and the adjacency algebras of finite association schemes. The definition of multiplicity generalizes that for schemes. For a broad class of these structures, which includes the adjacency algebras, it is proved that if all the nontrivial multiplicities are equal then the algebra is commutative, and is a C-algebra if its dimension is larger than two.

Association schemes, fusion rings, C-algebras, and reality-based algebras where all nontrivial multiplicities are equal