Homology representations of compactified configurations on graphs applied to $M_{2,n}$

We obtain new calculations of the top weight rational cohomology of the moduli spaces M2,n, equivalently the rational homology of the tropical moduli spaces ∆2,n, as a representation of Sn. These calculations are achieved fully for all n ≤ 10, and partially—for specific irreducible representations of Sn—for n ≤ 22. We also present conjectures, verified up to n = 22, for the multiplicities of the irreducible representations stdn and stdn ⊗ sgnn. We achieve our calculations via a comparison with the homology of compactified configuration spaces of graphs. These homology groups are equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. In this paper, we construct an efficient free resolution for these homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider irreducible representations individually, vastly simplifying the calculation of these homology representations from the free resolution.