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 M-2,M-n, equivalently the rational homology of the tropical moduli spaces Delta(2,n), as a representation of S-n. These calculations are achieved fully for all n <= 11, and partially-for specific irreducible representations of S-n-for n <= 22. We also present conjectures, verified up to n = 22, for the multiplicities of the irreducible representations std(n) and std(n) circle times sgn(n). 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, from which we extract calculations on irreducible representations one at a time, simplifying the calculation of these homology representations.