Geodesic
Association schemes, fusion rings, C-algebras, and reality-based algebras where all nontrivial multiplicities are equal
Journal of Algebraic Combinatorics, Tome 31 (2010) no. 4, pp. 491-499.

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Summary: 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.
Keywords: keywords reality-based algebra, C-algebra, adjacency algebra, association scheme, fusion ring, multiplicity 
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     author = {Blau, Harvey I.},
     title = {Association schemes, fusion rings, {C-algebras,} and reality-based algebras where all nontrivial multiplicities are equal},
     journal = {Journal of Algebraic Combinatorics},
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Blau, Harvey I. Association schemes, fusion rings, C-algebras, and reality-based algebras where all nontrivial multiplicities are equal. Journal of Algebraic Combinatorics, Tome 31 (2010) no. 4, pp. 491-499. http://geodesic.mathdoc.fr/item/JAC_2010__31_4_a4/