Characterization of table algebras by their multiplicities
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The structure of a standard integral table algebra can sometimes be determined by the multiplicities of the irreducible characters for that table algebra. In particular, it has been shown that a standard table algebra whose multiplicities are all equal (except for the trivial one) must be commutative. This dissertation seeks to extend the technique of using multiplicities to determine the structure of standard table algebras. For a commutative standard table algebra, we show that there exists exactly one character with nontrivial multiplicity if and only if the table basis is the wreath product of a two-dimensional subalgebra and an abelian group. Additionally, we show that a noncommutative standard integral table algebra with exactly one character (of degree two) that has nontrivial multiplicity must have one of two structures, both corresponding to a partial wreath product (B, D, C), where D has dimension 6 or 8, where C has dimension 2 or 3, and where the structure constants are explicitly determined by certain parameters that are bounded by a function of the nontrivial multiplicity.