Association schemes, fusion rings, C-algebras, and reality-based algebras where all nontrivial multiplicities are equal
Harvey I. Blau
DOI: 10.1007/s10801-009-0197-9
Abstract
Multiplicities corresponding to irreducible characters are defined for reality-based algebras. These algebras with a distinguished basis include fusion rings, C-algebras, and the adjacency algebras of finite association schemes. The definition of multiplicity generalizes that for schemes. For a broad class of these structures, which includes the adjacency algebras, it is proved that if all the nontrivial multiplicities are equal then the algebra is commutative, and is a C-algebra if its dimension is larger than two.
Pages: 491–499
Keywords: keywords reality-based algebra; C-algebra; adjacency algebra; association scheme; fusion ring; multiplicity
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References
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2. Arad, Z., Fisman, E., Muzychuk, M.: Generalized table algebras. Israel J. Math. 114, 29-60 (1999)
3. Bagherian, J., Barghi, A.R.: Standard character condition for C-algebras, [Math. RT] (17 March 2008)
4. Bannai, E., Ito, T.: Algebraic Combinatorics I. Association Schemes. Benjamin/Cummings, Menlo Park (1984)
5. Blau, H.I.: Quotient structures in C-algebras. J. Algebra 175, 24-64 (1995)
6. Blau, H.I.: Table algebras. European J. Combin. 30, 1426-1455 (2009)
7. Blau, H.I., Zieschang, P.-H.: Sylow theory for table algebras, fusion rule algebras, and hypergroups. J. Algebra 273, 551-570 (2004)
8. Evdokimov, S.A., Ponomorenko, I.N., Vershik, A.M.: Algebras in Plancherel duality and algebraic combinatorics. Funct. Anal. Appl. 31, 252-261 (1997)
9. Gelaki, S., Nikshych, D.: Nilpotent fusion categories. Adv. Math. 217, 1053-1071 (2008)
10. Green, R.M.: Tabular algebras and their asymptotic versions. J. Algebra 252, 27-64 (2002)
11. Hanaki, A., Uno, K.: Algebraic structure of association schemes of prime order. J. Algebraic Combin. 23, 189-195 (2006)
12. Higman, D.G.: Coherent algebras. Linear Algebra Appl. 93, 209-239 (1987)
13. Kawada, Y.: Über den Dualitätssatz der Charaktere nicht commutativen Gruppen. Proc. Phys. Math.
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