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Residually thin and nilpotent table algebras, which are abstractions of fusion rings and adjacency algebras of association schemes, are defined and investigated. A formula for the degrees of basis elements in residually thin table algebras is established, which yields an integrality result of Gelaki and Nikshych as an immediate corollary; and it is shown that this formula holds only for such algebras. These theorems for table algebras specialize to new results for association schemes.doi:10.5802/alco.194 fatcat:dgpbxvaoufc5jcvzeo7mwe5luu