Functional equations for orbifold wreath products

TL;DR

This paper derives generating functions for multiplicative invariants of orbifold wreath products, extending known formulas from finite group quotients to all closed, effective orbifolds, with applications to Euler characteristics.

Contribution

It generalizes product formulas for orbifold invariants from finite group quotients to all closed, effective orbifolds, including new combinatorial methods.

Findings

Derived generating functions for orbifold wreath products

Computed Euler and Euler--Satake characteristics for weighted projective spaces

Extended known formulas from finite group quotients to general orbifolds

Abstract

We present generating functions for extensions of multiplicative invariants of wreath symmetric products of orbifolds presented as the quotient by the locally free action of a compact, connected Lie group in terms of orbifold sector decompositions. Particularly interesting instances of these product formulas occur for the Euler and Euler--Satake characteristics, which we compute for a class of weighted projective spaces. This generalizes results known for global quotients by finite groups to all closed, effective orbifolds. We also describe a combinatorial approach to extensions of multiplicative invariants using decomposable functors that recovers the formula for the Euler--Satake characteristic of a wreath product of a global quotient orbifold.