Grothendieck Ring of Varieties with Actions of Finite Groups

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 cor...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society 2019-11, Vol.62 (4), p.925-948, Article 925
Main Authors: Gusein-Zade, S.M., Luengo, I., Melle-Hernández, A.
Format: Article
Language:English
Subjects:
Homomorphisms
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Summary: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.
ISSN:0013-0915
1464-3839
DOI:10.1017/S001309151900004X