Commutative Standard Table Algebras with at Most One Nontrivial Multiplicity

Abstract

We classify commutative standard table algebras (STA) with at most one nontrivial multiplicity. The main result shows that there exists exactly one nontrivial multiplicity if and only if the table basis is the wreath product of a two-dimensional subalgebra and an abelian group. The theorem applies to adjacency algebras of commutative association schemes with exactly one primitive idempotent matrix of rank greater than one. A theorem of Seitz that characterizes finite groups with exactly one irreducible representation of degree greater than one is another corollary of the main theorem.

Key Words:

• Association schemes
• Finite groups
• Irreducible characters
• Multiplicities
• Table algebras
• Wreath product

2010 Mathematics Subject Classification:

• 16W10
• 20C15
• 16K20
• 05E30

ACKNOWLEDGMENT

The material included in this paper comprises a chapter of the author's doctoral dissertation at Northern Illinois University, directed by Harvey I. Blau. The author would like to extend her deepest gratitude toward Dr. Blau for his invaluable suggestions, guidance, and support, in the development and the writing of this paper.

Notes

Communicated by M. Cohen.