We define a Grothendieck ring of varieties with actions of finite groups and show that the orbifold Euler characteristic and the Euler characteristics of higher orders can be defined as homomorphisms from this ring to the ring of integers. We describe two natural λ-structures on the ring and the corresponding power structures over it and show that one of these power structures is effective. We define a Grothendieck ring of varieties with equivariant vector bundles and show that the generalized (‘motivic’) Euler characteristics of higher orders can be defined as homomorphisms from this ring to the Grothendieck ring of varieties extended by powers of the class of the complex affine line. We give an analogue of the Macdonald type formula for the generating series of the generalized higher-order Euler characteristics of wreath products.

中文翻译：

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具有有限群作用的格洛腾迪克环

我们定义了具有有限群作用的簇的格洛腾迪克环，并证明了轨道欧拉特征和高阶欧拉特征可以定义为从这个环到整数环的同态。我们描述两个自然λ-环上的结构和相应的权力结构，并表明这些权力结构之一是有效的。我们定义了等变向量丛的格洛腾迪克环，并表明高阶的广义（“动机”）欧拉特征可以定义为从这个环到由复仿射类的幂扩展的格洛腾迪克环的同态线。我们给出了一个 Macdonald 型公式的类似物，用于生成环积的广义高阶 Euler 特征序列。