F-signature function of quotient singularities

TL;DR

This paper investigates the F-signature function of quotient singularities, demonstrating it is a quasi-polynomial with specific coefficient properties, and provides explicit formulas especially for cyclic groups.

Contribution

It establishes the quasi-polynomial nature of the F-signature function and characterizes its coefficients in terms of group invariants, with explicit formulas for cyclic groups.

Findings

F-signature function is a quasi-polynomial for quotient singularities.

The second coefficient of the quasi-polynomial is always zero.

Explicit formulas for coefficients when the acting group is cyclic.

Abstract

We study the shape of the F-signature function of a $d$-dimensional quotient singularity $\mathbb{k}[[ x_1,\ldots,x_d]]^G$, and we show that it is a quasi-polynomial. We prove that the second coefficient is always zero and we describe the other coefficients in terms of invariants of the finite acting group $G\subseteq \mathrm{Gl}(d,\Bbbk)$. When $G$ is cyclic, we obtain more specific formulas for the coefficients of the quasi-polynomial, which allow us to compute the general form of the function in several examples.