On \(S\)-matrix, and fusion rules for irreducible \(V^G\)-modules

Let \(V\) be a simple vertex operator algebra, and \(G\) a finite automorphism group of \(V\) such that \(V^G\) is regular. The definition of entries in \(S\)-matrix on \(V^G\) is discussed, and then is extended. The set of \(V^G\)-modules can be considered as a unitary space. In this paper, we obtain some connections between \(V\)-modules and \(V^G\)-modules over that unitary space. As an application, we determine the fusion rules for irreducible \(V^G\)-modules which occur as submodules of irreducible \(V\)-modules by the fusion rules for irreducible \(V\)-modules and by the structure of \(G\).