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

Abstract: The moduli space $\overline{\mathcal{M}}<em>{0,n}$ 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 $\overline{\mathcal{M}}</em>{0,n}$. This is achieved by studying wall crossings of the moduli spaces of quasimaps which provide us with a new inductive construction of $\overline{\mathcal{M}}<em>{0,n}$, equivariant with respect to the symmetric group action. Moreover we prove that $H^{{2k}(\overline{\mathcal{M}}}</em>{0,n})$ for $k\le 3$ and $$H^{{2k}(\overline{\mathcal{M}}_{0,n})\oplus} H<sup>{2k-2}(\overline{\mathcal{M}}_{0,n})$$ 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 $\mathbb{P}<sup>1[n]$ of the configuration space of $n$ points on $\mathbb{P}<sup>1$ and more generally on the cohomology of the moduli space $$\overline{\mathcal{M}}_{0,n}(\mathbb{P}<sup>{m-1},1)$$ of stable maps.