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.
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Keywords: Table algebra, fusion ring, association scheme, residually thin, nilpotent, thin central chain.
Blau, Harvey I. 1
CC-BY 4.0
@article{ALCO_2022__5_1_21_0,
author = {Blau, Harvey I.},
title = {On residually thin and nilpotent table algebras, fusion rings, and association schemes},
journal = {Algebraic Combinatorics},
pages = {21--36},
year = {2022},
publisher = {MathOA foundation},
volume = {5},
number = {1},
doi = {10.5802/alco.194},
language = {en},
url = {https://www.numdam.org/articles/10.5802/alco.194/}
}
TY - JOUR AU - Blau, Harvey I. TI - On residually thin and nilpotent table algebras, fusion rings, and association schemes JO - Algebraic Combinatorics PY - 2022 SP - 21 EP - 36 VL - 5 IS - 1 PB - MathOA foundation UR - https://www.numdam.org/articles/10.5802/alco.194/ DO - 10.5802/alco.194 LA - en ID - ALCO_2022__5_1_21_0 ER -
%0 Journal Article %A Blau, Harvey I. %T On residually thin and nilpotent table algebras, fusion rings, and association schemes %J Algebraic Combinatorics %D 2022 %P 21-36 %V 5 %N 1 %I MathOA foundation %U https://www.numdam.org/articles/10.5802/alco.194/ %R 10.5802/alco.194 %G en %F ALCO_2022__5_1_21_0
Blau, Harvey I. On residually thin and nilpotent table algebras, fusion rings, and association schemes. Algebraic Combinatorics, Tome 5 (2022) no. 1, pp. 21-36. doi: 10.5802/alco.194
[1] Generalized table algebras, Israel J. Math., Volume 114 (1999), pp. 29-60 | Zbl | MR | DOI
[2] Quotient structures in -algebras, J. Algebra, Volume 177 (1995) no. 1, pp. 297-337 | Zbl | MR | DOI
[3] Table algebras, European J. Combin., Volume 30 (2009) no. 6, pp. 1426-1455 | Zbl | MR | DOI
[4] Fusion rings with few degrees, J. Algebra, Volume 396 (2013), pp. 220-271 | Zbl | MR | DOI
[5] Normal series and character values in -standard table algebras, Comm. Algebra, Volume 45 (2017) no. 11, pp. 4646-4655 | Zbl | MR | DOI
[6] Sylow theory for table algebras, fusion rule algebras, and hypergroups, J. Algebra, Volume 273 (2004) no. 2, pp. 551-570 | Zbl | MR | DOI
[7] On residually thin hypergroups, J. Algebra, Volume 551 (2020), pp. 93-118 | Zbl | MR | DOI
[8] Nilpotent fusion categories, Adv. Math., Volume 217 (2008) no. 3, pp. 1053-1071 | Zbl | MR | DOI
[9] Nilpotent schemes and group-like schemes, J. Combin. Theory Ser. A, Volume 115 (2008) no. 2, pp. 226-236 | Zbl | MR | DOI
[10] Classification of association schemes with small vertices (2021) (Accessed 2016-08-19 http://math.shinshu-u.ac.jp/~hanaki/as/)
[11] Indecomposable decompositions of modular standard modules for two families of association schemes, J. Algebraic Combin., Volume 46 (2017) no. 2, pp. 445-453 | Zbl | MR | DOI
[12] A wedge product of association schemes, European J. Combin., Volume 30 (2009) no. 3, pp. 705-715 | Zbl | MR | DOI
[13] An algebraic approach to association schemes, Lecture Notes in Mathematics, 1628, Springer-Verlag, Berlin, 1996, xii+189 pages | Zbl | MR | DOI
[14] Theory of association schemes, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005, xvi+283 pages | MR
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