A Frobenius formula for the structure coefficients of double-class algebras of Gelfand pairs

ABSTRACT

After a careful consideration of some of the well known properties of irreducible characters of finite groups to zonal spherical functions of Gelfand pairs, we were able to deduce a Frobenius formula for Gelfand pairs. For a given Gelfand pair, the structure coefficients of its associated double-class algebra can be written in terms of zonal spherical functions. This is a generalization of the Frobenius formula which expresses the structure coefficients of the center of a finite group algebra in terms of irreducible characters.

KEYWORDS:

• Double-class algebras
• Gelfand pairs and zonal spherical functions
• group algebras
• representation theory of finite groups
• structure coefficients

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 05E15
• 20C08

Acknowledgment

This work was done in the Laboratoire Bordelais de Recherche en Informatique at the University of Bordeaux while I was a Ph.D. student under the supervision of Jean-Christophe Aval and Valentin Fray. I would like to extend a warm thanks for all their encouragements and suggestions to develop this generalization of the Frobenius theorem.

Notes

¹¹ x∖c is the permutation obtained from x by removing the cycle c. The definition of cycle-type can be extented naturally to this kind of permutations.