Generalized Commutative Association Schemes, Hypergroups, and Positive Product Formulas

TL;DR

This paper generalizes finite commutative association schemes to broader discrete cases, showing they lead to hypergroups with positive dual convolutions and suggesting potential for new dual positive product formulas.

Contribution

It introduces new discrete generalizations of association schemes that produce hypergroups with positive dual convolutions, expanding the theoretical framework.

Findings

Discrete hypergroups admit dual positive convolutions.

Generalized schemes lead to hypergroups with positive dual structures.

Potential for discovering new dual positive product formulas.

Abstract

It is well known that finite commutative association schemes in the sense of the monograph of Bannai and Ito lead to finite commutative hypergroups with positive dual convolutions and even dual hypergroup structures. In this paper we present several discrete generalizations of association schemes which also lead to associated hypergroups. We show that discrete commutative hypergroups associated with such generalized association schemes admit dual positive convolutions at least on the support of the Plancherel measure. We hope that examples for this theory will lead to the existence of new dual positive product formulas in near future.