Bangteng Xu

Table algebras with exactly one irreducible character whose degree and multiplicity are not equal

ABSTRACT Commutative standard table algebras with exactly one multiplicity not equal to 1 are characterized by the wreath product of some special table algebras in [1]. A natural and much more general question is the characterization of standard table algebras (not necessarily commutative) with exactly one irreducible character whose degree and multiplicity are not equal and the degree is 1. We will give a characterization of such table algebras, including the main result of [1] as a special case. Applications to association schemes are also discussed.

Combinatorial extensions of Terwilliger algebras and wreath products of association schemes

Abstract We introduce the notion of the combinatorial extension of a Terwilliger algebra by a coherent algebra. By using this notion, we find a simple way to describe the Terwilliger algebras of certain coherent configurations as combinatorial extensions of simpler Terwilliger algebras. In particular, given an association scheme S and another association scheme R such that the Terwilliger algebra of R is isomorphic to a coherent algebra, we prove that the Terwilliger algebra of the wreath product S ≀ R is isomorphic to the combinatorial extension of the Terwilliger algebra of S by a coherent algebra. We also show that the Terwilliger algebra of the wreath product W of rank 2 association schemes can be expressed as the combinatorial extension of adjacency algebras of association schemes induced by the closed subsets of W . As a direct consequence, we obtain simple conceptual explanations and alternative proofs of many known results on the structures of Terwilliger algebras of wreath products of association schemes.

On Isomorphisms Between Integral Table Algebras and Applications to Finite Groups and Association Schemes

A problem that has been open for a long time is whether an automorphism of the center of the integral group ring of a finite group permutes the class sums. As a generalization, the similar problem for integral table algebras has also been studied in a few papers. In this paper we further the investigation of the problem for integral C-algebras and table algebras. We will first present some necessary and sufficient conditions under which an algebra isomorphism between integral C-algebras is monomial, and as an application, obtain a conceptual proof of the class sum correspondence theorem of group rings of finite groups. Then we prove that an algebra isomorphism compatible with degree maps between standard integral nilpotent table algebras over the ring of algebraic integers is exact. As a direct consequence, we get Hertwecks result [6] for finite nilpotent groups. An application to association schemes is also presented.

On P-Polynomial Table Algebras and Applications to Association Schemes

By investigating the primitive idempotents of a commutative table algebra that has a table basis element with all distinct eigenvalues, we prove a necessary and sufficient condition in terms of the eigenvalues of a table basis element b ∈ B under which a real table algebra (A, B) is a P-polynomial table algebra. As direct consequences, we obtain the necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial association scheme in [7] as well as a necessary and sufficient condition for a symmetric association scheme to be a P-polynomial association scheme.

On Perfect P-polynomial Table Algebras

The Bose–Mesner algebra of the association scheme of the ordinary n-gon has the following remarkable properties: (i) It has a P-polynomial structure with respect to every faithful basis element; and (ii) Any closed subset generated by a basis element has a P-polynomial structure with respect to this basis element. C-algebras or table algebras that have these two properties are called perfect P-polynomial C-algebras or table algebras. By applying and extending some of the techniques developed in Xu (2006), we will give a classification of perfect P-polynomial table algebras in terms of intersection matrices. As a direct consequence of this classification, we will prove that a standard real integral table algebra (A, B) with |B| ≥ 6 is a perfect P-polynomial table algebra if and only if it is exactly isomorphic to the Bose–Mesner algebra of the association scheme of the ordinary (2|B|−2)-gon or (2|B|−1)-gon. This result generalizes part of the main theorem in Xu (2006). We will present examples revealing that this result is not true if |B| ≤ 5.

Noncommutative reality-based algebras of rank 6

ABSTRACT We show that noncommutative standard reality-based algebras (RBAs) of dimension 6 are determined up to exact isomorphism by their character tables. We show that the possible character tables of these RBAs are determined by seven real numbers, the first four of which are positive and the remaining three real numbers can be arbitrarily chosen up to a single exception. We show how to obtain a concrete matrix realization of the elements of the RBA-basis from the character table. Using a computer implementation, we give a list of all noncommutative integral table algebras of rank 6 with orders up to 150. Four in the list are primitive, but we show three of them cannot be realized as adjacency algebras of association schemes. In the last section of the paper, we apply our methods to give a precise description of the noncommutative integral table algebras of rank 6 for which the multiplicity of both linear characters is 1.

Table algebras with central fusions

ABSTRACT The character theory of table algebras is not as good as the character theory of finite groups. We introduce the notion of a table algebra with a central-fusion, in which the character theory has better properties. We study conditions under which a table algebra (A,B) has a central-fusion, and its central-fusion is exactly isomorphic to the wreath product of the central-fusion of a quotient table algebra of (A,B) and another table algebra. As a consequence, we obtain a complete characterization of table algebras with exactly one irreducible character whose degree and multiplicity are not equal. Applications to association schemes are also discussed.

Structures of commutative table algebras determined by their character tables and applications to finite groups

ABSTRACT Structures of table algebras determined by their character tables are studied in [6]. In this paper we continue the research in this direction. In particular, we investigate the conditions under which the character table with a zero submatrix yields a generalized wreath product for a commutative table algebra. Applications to finite groups are also discussed, and some new and known results are obtained as direct consequences.

Investigations on association schemes with elementary abelian thin residue

Abstract In this paper, we give a new class of association schemes whose thin residues are isomorphic to an elementary abelian p -group of order p 2 . We then study the automorphism groups of these schemes and determine whether these schemes are schurian.

On a class of nilpotent table algebras and applications to association schemes

In this paper we present sufficient conditions under which a nilpotent table algebra is exactly isomorphic to the wreath product of thin table algebras, and show by examples that this result is not true when the conditions are replaced by weaker ones. Applications to association schemes are also discussed.