# Moscow Mathematical Journal

Volume 16, Issue 4, October–December 2016 pp. 751–765.

Equivariant Versions of Higher Order Orbifold Euler Characteristics

**Authors**:
S. M. Gusein-Zade (1), I. Luengo (2) and A. Melle-Hernández (2)

**Author institution:**(1) Moscow State University, Faculty
of Mathematics and Mechanics, GSP-1, Moscow, 119991, Russia

(2) ICMAT (CSIC-UAM-UC3M-UCM); Complutense University of Madrid, Dept. of Algebra, Madrid, 28040, Spain

**Summary: **

There are (at least) two different approaches to define an equivariant analogue of the Euler characteristic for a space with a finite group action. The first one defines it as an element of the Burnside ring of the group. The second approach emerged from physics and includes the orbifold Euler characteristic and its higher order versions. Here we give a way to merge the two approaches together defining (in a certain setting) higher order Euler characteristics with values in the Burnside ring of a group. We give Macdonald type equations for these invariants. We also offer generalized (“motivic”) versions of these invariants and formulate Macdonald type equations for them as well.

2010 Math. Subj. Class. 55M35, 32Q55, 19A22.

**Keywords:**Finite group actions, orbifold Euler characteristic, Burnside ring, complex quasi-projective varieties, wreath products, generating series.

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