  It is known that a certain invariant subring $R$ has finite
$F$-representation type. Thus, we can write the $R$-module ${}^eR$ as a finite
direct sum of finitely many $R$-modules. In such a decomposition of ${}^eR$, we
pay attention to the multiplicity of each direct summand. For the multiplicity
of free direct summand, there is the notion of $F$-signature defined by C.
Huneke and G. Leuschke and it characterizes some singularities. In this paper,
we extend this notion to non free direct summands and determine the explicit
values of them.

Commutative Algebra;Representation Theory