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

Document Type

Article

Publication Date

1-1-2024

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

Publication Source (Journal or Book title)

Experimental Mathematics

First Page

611

Last Page

623

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