# ALGEBRAIC COMBINATORICS

On residually thin and nilpotent table algebras, fusion rings, and association schemes
Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 21-36.

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. Bi-anchored 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.

Revised:
Accepted:
Published online:
DOI: 10.5802/alco.194
Classification: 16P10,  16P70,  16W10,  20D15,  05E30
Keywords: Table algebra, fusion ring, association scheme, residually thin, nilpotent, thin central chain.
Blau, Harvey I. 1

1 Department of Mathematical Sciences Northern Illinois University DeKalb, IL 60115, USA
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Blau, Harvey I. On residually thin and nilpotent table algebras, fusion rings, and association schemes. Algebraic Combinatorics, Volume 5 (2022) no. 1, pp. 21-36. doi : 10.5802/alco.194. https://alco.centre-mersenne.org/articles/10.5802/alco.194/

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