Abstract
If the distinguished basis of a table algebra is an irredundant union of n proper closed subsets, and if the positive structure constants of the quotient table algebra (rescaled to be standard) modulo the intersection of these closed subsets are all at least 1, then it is proved that the order of this quotient algebra is bounded above by a function of n. This generalizes a result of B. H. Neumann for finite groups, applies directly to association schemes, and also yields the following result: if G is a finite group, 𝒦 is the set of minimal members (with respect to containment) of the set of kernels of irreducible characters of G, and N = ∏ K∈𝒦 K, then |N| is bounded above by a function of |𝒦|. Table algebras where the table basis is a union of three or four proper closed subsets are characterized as well.
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Presented by Anatoly Vershik.
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Blau, H.I., Chen, G. Table Bases as Unions of Proper Closed Subsets. Algebr Represent Theor 17, 1527–1552 (2014). https://doi.org/10.1007/s10468-013-9458-3
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DOI: https://doi.org/10.1007/s10468-013-9458-3