Generalized F-signature of invariant subrings

It is known that a certain invariant subring R has finite F-representation type. Thus, we can write the R  -module Re as a finite direct sum of finitely many R  -modules. In such a decomposition of Re, 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 their explicit values.