Duality of translation association schemes coming from certain actions

TL;DR

This paper explores the duality properties of translation association schemes derived from finite group actions on abelian groups, revealing conditions for self-duality and providing numerous examples including Hamming and sesquilinear forms schemes.

Contribution

It introduces a new framework linking group actions to the duality of association schemes, generalizing previous constructions and illustrating with diverse examples.

Findings

Self-duality of schemes under certain group map conditions

Extension of duality to schemes from two group actions

Examples include Hamming, sesquilinear forms, and weak Hamming schemes

Abstract

Translation association schemes are constructed from actions of finite groups on finite abelian groups satisfying certain natural conditions. It is also shown that the mere existence of maps from finite groups to themselves sending each element in their groups to its `adjoint' entails the self-duality of the constructed association schemes. Many examples of these, including Hamming scheme and sesquilinear forms schemes, are provided. This con- struction is further generalized to show the duality of the association schemes coming from actions of two finite groups on the same finite abelian group. An example of this is supplied with weak Hamming schemes.