# Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups

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## Abstract

**:**

## 1. Introduction

## 2. Power Structures and the Grothendieck Ring of Varieties with Actions of Finite Groups

- (1)
- ${\left(A\left(t\right)\right)}^{0}=1$;
- (2)
- ${\left(A\left(t\right)\right)}^{1}=A\left(t\right)$;
- (3)
- ${\left(A\left(t\right)\xb7B\left(t\right)\right)}^{m}={\left(A\left(t\right)\right)}^{m}\xb7{\left(B\left(t\right)\right)}^{m}$;
- (4)
- ${\left(A\left(t\right)\right)}^{m+n}={\left(A\left(t\right)\right)}^{m}\xb7{\left(A\left(t\right)\right)}^{n}$;
- (5)
- ${\left(A\left(t\right)\right)}^{mn}={\left({\left(A\left(t\right)\right)}^{n}\right)}^{m}$;
- (6)
- ${(1+{a}_{1}t+\dots )}^{m}=1+m{a}_{1}t+\dots $;
- (7)
- ${\left(A\left({t}^{k}\right)\right)}^{m}={\left(A\left(t\right)\right)}^{m}|{}_{t\mapsto {t}^{k}}$ for $k\in {\mathbb{Z}}_{>0}$.

**Definition**

**1.**

- (1)
- if$(X,G)$and$({X}^{\prime},{G}^{\prime})$are isomorphic, then$\left[(X,G)\right]=\left[({X}^{\prime},{G}^{\prime})\right]$;
- (2)
- if Y is a Zariski closed G-subvariety of a G-variety X, then$\left[\right(X,G\left)\right]=\left[\right(Y,G\left)\right]+\left[\right(X\backslash Y,G\left)\right]$;
- (3)
- if$(X,G)$is a G-variety and G is a subgroup of a finite group H, then$\left[({\mathrm{Ind}}_{G}^{H}X,H)\right]=\left[(X,G)\right]$.

**Remark**

**1.**

**Definition**

**2.**

## 3. The Universal Euler Characteristic

**Remark**

**2.**

## 4. Macdonald-Type Equations and $\mathbf{\lambda}$-Structure Homomorphisms

**Theorem**

**1.**

**Remark**

**3.**

## 5. Generalized Euler Characteristics of Higher Orders as Homomorphisms from ${\mathbf{K}}_{\mathbf{0}}^{\mathrm{fGr}}\left({\mathrm{Var}}_{\mathbb{C}}\right)$

**Definition**

**3.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**2.**

**Remark**

**4.**

## 6. A Substitute of a Macdonald-Type Equation for the Homomorphism $\mathbf{\alpha}$

**Theorem**

**3.**

**Proof.**

**Corollary**

**1.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Gusein-Zade, S.M.; Luengo, I.; Melle-Hernández, A.
Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups. *Symmetry* **2019**, *11*, 902.
https://doi.org/10.3390/sym11070902

**AMA Style**

Gusein-Zade SM, Luengo I, Melle-Hernández A.
Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups. *Symmetry*. 2019; 11(7):902.
https://doi.org/10.3390/sym11070902

**Chicago/Turabian Style**

Gusein-Zade, S. M., I. Luengo, and A. Melle-Hernández.
2019. "Generalized Orbifold Euler Characteristics on the Grothendieck Ring of Varieties with Actions of Finite Groups" *Symmetry* 11, no. 7: 902.
https://doi.org/10.3390/sym11070902