Table algebras and association schemes with elementary abelian thin residues
Gang Chen, Sheng-an Chen, Bangteng Xu · Communications in Algebra · 2025
Association schemes in which the thin residues are thin have been studied in many papers. A class of association schemes with elementary abelian thin residues have played an important role in the study of some p-schemes and p-valenced schemes. In the study of association schemes with thin thin residues, the existence, constructions, and schurity problems are among the major research topics. In this paper we continue the research in [Citation4, Citation11]. We will first study the existence and constructions of table algebras with elementary abelian thin residues, and use them to construct association schemes that also have elementary abelian thin residues. We will show that such table algebras with the minimum dimensions are exactly isomorphic to each other, and such association schemes with the minimum ranks are separable. The schurity problems of the association schemes are also discussed. In particular, we will prove that a class of commutative association schemes with elementary abelian thin residues are not schurian.