P\'olya enumeration, wreath product symmetric functions, and moduli spaces of curves

TL;DR

This paper introduces a new algebraic framework combining Pólya enumeration and wreath product symmetric functions to compute $S_n$-equivariant Euler characteristics of moduli spaces of curves, simplifying complex calculations.

Contribution

It develops an enriched algebraic calculus for equivariant Euler characteristics using wreath product symmetric functions and connects it to polynomial functors and Grothendieck rings.

Findings

Derived explicit formulas for generating functions of equivariant Euler characteristics.

Established an action of wreath product symmetric functions on ordinary symmetric functions.

Connected combinatorial enumeration with algebraic structures like polynomial functors.

Abstract

We develop a calculus for $S_n$-equivariant Euler characteristics of moduli spaces of stable curves and stable maps. Our approach involves an enrichment of P\'olya's cycle index polynomial of a graph to a certain algebra $\Lambda^{[2]}$ of wreath product symmetric functions. Building on foundational work of Macdonald, we prove that $\Lambda^{[2]}$ may be viewed as the Grothendieck ring of the category of polynomial functors which map symmetric sequences of vector spaces to vector spaces. This interpretation gives rise to an action of $\Lambda^{[2]}$ on the ordinary ring of symmetric functions $\Lambda$, which is described concretely in terms of Adams operations and skewing by power sums. This action lets us deduce appealing formulas, involving only ordinary symmetric functions, for generating functions of $S_n$-equivariant Euler characteristics.