On residually thin and nilpotent table algebras, fusion rings, and association schemes

Residually thin and nilpotent table algebras, which are abstractions of fusion rings adjacency algebras association schemes, defined investigated. A formula for the degrees basis elements in residually is established, yields an integrality result Gelaki Nikshych as immediate corollary; it shown that this holds only such algebras. These theorems specialize to new results schemes. Bi-anchored thin-central (BTC) chains closed subsets used define nilpotence, manner Hanaki Lower BTC-chains abstraction lower central series a finite group. partial characterization proved; family examples illustrates unlike case groups, there not necessarily unique BTC-chain algebra or scheme.