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 M 2 ,n , equivalently the rational homology of the tropical moduli spaces ∆ 2 ,n , as a representation of S n . These calculations are achieved fully for all n ≤ 10, 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 ⊗ 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 con-struct 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.