Generalized F-signature of invariant subrings

TL;DR

This paper extends the concept of F-signature to include non-free summands in invariant subrings with finite F-representation type, providing explicit calculations for these generalized signatures.

Contribution

It introduces a generalized F-signature concept for all summands in invariant subrings, expanding the understanding of singularities beyond free modules.

Findings

Explicit values of generalized F-signature for invariant subrings

Extension of F-signature to non-free summands

Characterization of singularities using generalized signatures

Abstract

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.