Homology Representations of Compactified Configurations on Graphs Applied to 𝓜2,n

Abstract

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 , as a representation of Sn . These calculations are achieved fully for all , and partially—for specific irreducible representations of Sn —for . We also present conjectures, verified up to n = 22, for the multiplicities of the irreducible representations and . 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.