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The classification is here completed for the p -standard table algebras of order p 3 for any rational prime p. These include the adjacency algebras of p -schemes of order p 3. The remaining case, where the thin radical of the distinguished basis of the table algebra has order p 2 , is resolved. The algebras in this case are explicitly determined up to isomorphism as wreath products, partial wreath products, or as members of a more complex family called hexagonal standard table algebras. These are parametrized by certain correspondences introduced in this article called hexagonal functions , that are defined in general from a subset of a group to an arbitrary set. [ABSTRACT FROM AUTHOR]

A positive integer n H (the valency) is assigned to any residually thin finite hypergroup H ; and for any set π of prime numbers, π -valency is defined for elements of H. These notions coincide with those of valency (resp. order and degree) when H appears as the relations of an association scheme (resp. the distinguished basis of a standard table algebra). When all elements of H are π -valenced, existence is established of a closed subset U (called the π - radical of H) such that n U is a π -number, U contains all subnormal closed subsets V with n V a π -number, and the quotient hypergroup H / / U is a group. Then the classical results on existence and conjugacy of Hall π -subgroups for H / / U when the group is solvable (or more generally π -separable), and Sylow theory for the group when π is a singleton set, are extended to the Hall π -subsets of the hypergroup H. Thus, Vasil'ev and Zieschang's results on solvable association schemes are generalized. [ABSTRACT FROM AUTHOR]

The p -standard table algebras of order p 3 for any rational prime p , which include the adjacency algebras of p -schemes of order p 3 , are classified, provided that the thin radical of the distinguished basis of the table algebra has order p. The classification is in terms of wreath products, and of certain vectors (that we call Legendre elements) in the group algebra of a group of order p. Their coefficients yield real solutions to the classical equations that hold for the Legendre symbols. A fully explicit classification of the relevant Legendre elements, and hence of the table algebras at issue, is attained in the case of integer structure constants, as we characterize certain real solutions to these equations sufficiently enough to show that the only integer solutions are in fact ± the Legendre symbols themselves and appropriate translates thereof. Criteria for two such table algebras being isomorphic are also established in this case, and the automorphism group of any one such table algebra is determined. [ABSTRACT FROM AUTHOR]

It is shown that a two-dimensional irreducible representation of a reality-based algebra with a standard degree map is determined by the other irreducible representations of the algebra and the action of its given anti-automorphism on the distinguished basis. This generalizes known results for algebras of dimensions 5, 6, 7, and 8. Explicit conclusions equivalent to those for dimensions 5 and 6 are recovered here by specializing the procedure of proof of the main theorem. [ABSTRACT FROM AUTHOR]

Homomorphisms and quotient structures of reality-based algebras are presented and described; related central idempotents and quotient sets are characterized; and the extent to which a homomorphism preserves the given anti-automorphisms of the algebras is determined. The notion of a partial wreath product, an algebraic abstraction of the wedge product of association schemes, is developed. Prototypical results previously obtained for various special cases (including table algebras and commutative C -algebras) are thus uniformly generalized. [ABSTRACT FROM AUTHOR]

Abstract Standard table algebras that have an irreducible character of degree two, and with all other irreducible characters of multiplicity one, are classified to exact isomorphism. [ABSTRACT FROM AUTHOR]

Abstract: Wreath products in reality-based algebras are generalizations of wreath products of table algebras and generalized Camina–Frobenius pairs of C-algebras. In this paper we present characterizations of the wreath product in a reality-based algebra by its irreducible characters and by the size of the zero submatrix of its character table. Applications to finite groups, table algebras, and association schemes are also discussed. In particular, we will show that the wreath product of one-class association schemes is characterized by the zeros in its first eigenmatrix. [Copyright &y& Elsevier]

Abstract: The main result characterizes fusion rings whose degree set consists of 1 and a fixed prime p. A well known theorem of Isaacs and Passman on finite groups whose irreducible character degrees are 1 and p follows as a corollary. Numerically compatible (NC) fusion rings are defined as a potential generalization of the Grothendieck rings of fusion categories; and the NC fusion rings of order p, , , pq, and (where p, q are primes) are classified in terms of certain standard table algebras, as another application of the main theorem. [Copyright &y& Elsevier]

Abstract: A lower bound on the number of elements outside a closed subset of a C-basis of a commutative C-algebra with dual a table algebra is derived, as is an equivalent condition for when the lower bound is met. As corollaries, lower bounds are obtained on the number of primitive idempotent matrices of rank greater than 1 in the adjacency algebra of a commutative, imprimitive association scheme; and, for a given normal subgroup N of a finite group G, on the number of irreducible characters of G whose kernels do not contain N, and on the number of conjugacy classes of G not contained in N. Also found are equivalent conditions for when these lower bounds are met. [Copyright &y& Elsevier]

Abstract: Two theorems on the decomposition of products in nilpotent commutative table algebras are established. This generalizes several results of Adan-Bante on the decomposition of products of irreducible characters and of conjugacy classes of finite p-groups, and extends these results to fusion rule algebras and association schemes. [Copyright &y& Elsevier]

Abstract: This article surveys table algebras as algebraic abstractions of association schemes and other structures. Various definitions are compared; some basic properties are developed; and some theorems and applications are presented. [Copyright &y& Elsevier]