Quotient structures in \(C\)-algebras (Q1895608)

A \(C\)-algebra is a commutative algebra \(A\) over the complex numbers with a distinguished basis \({\mathbf B}\) which has certain specified properties. In particular, the structure constants are real. The definition of a \(C\)-algebra was first stated explicitly by Kawada, following Hoheisel, although the notion is implicit in previous work of Schur. Kawada's goal was to abstract the relationship between the center of the group algebra and the ring of class functions for a finite group. Thus, for any \(C\)-algebra \((A,{\mathbf B})\), he constructed a dual \(C\)-algebra \((A,\widehat{\mathbf B})\), so that \(\widehat{\widehat{\mathbf B}}={\mathbf B}\), and so that when \(A\) is the center of a group algebra with basis \(\mathbf B\) consisting of the sums over conjugacy classes, then \((A,\widehat{\mathbf B})\) may be identified with the algebra of class functions with basis the set of irreducible characters of the group. The central aim of this paper is to clarify the relationship between substructures and quotient structures in \(C\)-algebras. Theorem 1 gives an explicit bijection between the quotient structures (resp. substructures) of an arbitrary \(C\)-algebra and the substructures (resp. quotient structures) of its dual. This extends a result of \textit{Z. Arad} and \textit{E. Fisman} [Commun. Algebra 19, No. 11, 2955-3009 (1991; Zbl 0790.20016), Theorem 2.9]. Theorem 2 then shows that one need assume only that \((A,{\mathbf B})\) (and not necessarily also its dual) is a table algebra in order to guarantee that any substructure of \((A,{\mathbf B})\) yields a quotient structure of \((A,{\mathbf B})\), and conversely. Then the Jordan-Hölder type theorem proved by Rao, Ray-Chaudhuri, and Singhi under Kawada's assumptions [\textit{E. Bannai}, \textit{T. Ito}, Algebraic combinatorics I, Theorem II.9.11]\ holds for all table algebras (Theorem 5), as does an analog (Theorem 6) of the Krull-Schmidt type theorem proved for commutative association schemes by \textit{P. Ferguson} and \textit{A. Turull} [J. Algebra 96, 211-229 (1985; Zbl 0573.20051), Theorem 3.17]. Theorems 7 and 8 generalize some results from group theory to suitable table algebras.