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

Abstract

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 Hertweck's result [6] for finite nilpotent groups. An application to association schemes is also presented.

Key Words:

• Algebra isomorphisms
• C-algebras
• Group rings
• Monomial isomorphisms
• Table algebras

2010 Mathematics Subject Classification:

• 20C05
• 20C15
• 13A99
• 05E30

ACKNOWLEDGMENTS

The author is sincerely grateful to H. Blau for carefully reading the paper and giving useful suggestions, especially the short proof of Lemma 4.2. The author would also like to thank the referee, whose comments improved the quality of the paper.

Notes

Communicated by A. Turull.