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.

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... d thin-central (BTC) chains of closed subsets are used to define nilpotence, in the manner of Hanaki for association schemes. Lower BTC-chains are defined as an abstraction of the lower central series of a finite group. A partial characterization is proved; and a family of examples illustrates that unlike the case for finite groups, there is not necessarily a unique lower BTC-chain for a nilpotent table algebra or association scheme.