Representations on the cohomology of $\overline{\mathcal{M}}_{0,n}$

TL;DR

This paper derives a closed formula for the symmetric group action on the cohomology of the moduli space of genus zero stable curves with marked points, revealing permutation representation structures and extending to related moduli spaces.

Contribution

It introduces a new inductive method via wall crossings of quasimaps to compute the symmetric group action on cohomology, providing explicit character formulas and permutation representation results.

Findings

Closed formula for $S_n$-action on $ar{M}_{0,n}$ cohomology.

Proves certain cohomology groups are permutation representations.

Extends methods to Fulton-MacPherson compactification and stable maps.

Abstract

The moduli space $\overline{\mathcal{M}}_{0,n}$ of $n$ pointed stable curves of genus $0$ admits an action of the symmetric group $S_n$ by permuting the marked points. We provide a closed formula for the character of the $S_n$-action on the cohomology of $\overline{\mathcal{M}}_{0,n}$. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of $\overline{\mathcal{M}}_{0,n}$, equivariant with respect to the symmetric group action. Moreover we prove that $H^{2k}(\overline{\mathcal{M}}_{0,n})$ for $k\le 3$ and $H^{2k}(\overline{\mathcal{M}}_{0,n})\oplus H^{2k-2}(\overline{\mathcal{M}}_{0,n})$ for any $k$ are permutation representations. Our method works for related moduli spaces as well and we provide a closed formula for the character of the $S_n$-representation on the cohomology of the Fulton-MacPherson compactification $\mathbb{P}^1[n]$ of the configuration space of $n$ points on $\mathbb{P}^1$ and more generally on the cohomology of the moduli space $\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{m-1},1)$ of stable maps.