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The moduli space $\overline{\mathcal{M}}_{0,n}$ of $n$ pointed stable curvesof genus $0$ admits an action of the symmetric group $S_n$ by permuting themarked points. We provide a closed formula for the character of the$S_n$-action on the cohomology of $\overline{\mathcal{M}}_{0,n}$. This isachieved by studying wall crossings of the moduli spaces of quasimaps whichprovide 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$H^{2k}(\overline{\mathcal{M}}_{0,n})\oplusH^{2k-2}(\overline{\mathcal{M}}_{0,n})$ for any $k$ are permutationrepresentations. Our method works for related moduli spaces as well and weprovide a closed formula for the character of the $S_n$-representation on thecohomology of the Fulton-MacPherson compactification $\mathbb{P}^1[n]$ of theconfiguration space of $n$ points on $\mathbb{P}^1$ and more generally on thecohomology of the moduli space$\overline{\mathcal{M}}_{0,n}(\mathbb{P}^{m-1},1)$ of stable maps.

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