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 $G$.