Duality of translation association schemes coming from certain actions
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Translation association schemes are constructed from ac-tions of ¯nite groups on ¯nite abelian groups satisfying certain natural conditions. It is also shown that the mere existence of maps from ¯-nite 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 construction is further generalized to show the duality of the association schemes coming from actions of two ¯nite groups on the same ¯nite abelian group. An example of this is supplied with weak Hamming schemes.
Abstract
Translation association schemes are constructed from ac-tions of ¯nite groups on ¯nite abelian groups satisfying certain natural conditions. It is also shown that the mere existence of maps from ¯-nite 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 construction is further generalized to show the duality of the association schemes coming from actions of two ¯nite groups on the same ¯nite abelian group. An example of this is supplied with weak Hamming schemes.
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