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Pólya enumeration, wreath product symmetric functions, and moduli spaces of curves

Siddarth Kannan, Terry Dekun Song

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 Pólya's cycle index polynomial of a graph to a certain algebra $Λ^{[2]}$ of wreath product symmetric functions. Building on foundational work of Macdonald, we prove that $Λ^{[2]}$ 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 $Λ^{[2]}$ on the ordinary ring of symmetric functions $Λ$, 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.

Pólya enumeration, wreath product symmetric functions, and moduli spaces of curves

Abstract

We develop a calculus for -equivariant Euler characteristics of moduli spaces of stable curves and stable maps. Our approach involves an enrichment of Pólya's cycle index polynomial of a graph to a certain algebra of wreath product symmetric functions. Building on foundational work of Macdonald, we prove that 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 on the ordinary ring of symmetric functions , 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 -equivariant Euler characteristics.
Paper Structure (8 sections, 9 theorems, 54 equations, 2 figures, 1 table)

This paper contains 8 sections, 9 theorems, 54 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. The coefficients $O(\Theta)$
  3. The moduli space of stable maps
  4. Wreath symmetric functions and Schur--Weyl duality
  5. $\mathbb{S}^{[2]}$-modules and polynomial functors of $\mathbb{S}$-modules
  6. The action of $\Lambda^{[2]}$ on $\Lambda$
  7. Representation-theoretic interpretation of the action
  8. The Pólya--Petersen character of a graph

Key Result

Theorem 1

We have where each coefficient $O(\Theta) \in \mathbb{Q}$ is the solution to a graph enumeration problem defined by (eqn:Otheta) in § subsec:graph_coeffs below.

Figures (2)

  • Figure 1: A point of $\overline{\mathcal{M}}_{5, 5}$ with dual graph given in Figure \ref{['fig:graph_exmp']}
  • Figure 2: The dual graph of the nodal algebraic curve in Figure \ref{['fig:stable_curve_exmp']}

Theorems & Definitions (14)

  • Theorem 1
  • Lemma 1: Specht
  • Theorem 2
  • Corollary 1
  • Definition 1
  • Definition 2
  • Theorem 3: Schur--Weyl duality for $\mathbb{S}$-modules
  • Definition 3
  • Theorem 4: Schur--Weyl duality for $\mathbb{S}^{[2]}$-modules
  • Lemma 2
  • ...and 4 more