Recursive algorithm and log-concavity of representations on the cohomology of $\overline{\mathcal M}_{0,n}$

TL;DR

This paper introduces a recursive algorithm to analyze the symmetric group representations on the cohomology of moduli spaces of genus 0 curves, revealing log-concavity properties and providing explicit formulas.

Contribution

It presents a new recursive method for computing these representations and establishes log-concavity of the Poincaré polynomial asymptotically.

Findings

Asymptotic log-concavity of the Poincaré polynomial.

Explicit formulas for invariant parts of cohomology.

Conjecture on equivariant log-concavity of cohomology modules.

Abstract

We provide a programmable recursive algorithm for the $\mathbb{S}_n$-representations on the cohomology of the moduli spaces $\overline{\mathcal M}_{0,n}$ of $n$-pointed stable curves of genus 0. As an application, we find explicit inductive and asymptotic formulas for the invariant part $H^*(\overline{\mathcal M}_{0,n}/\mathbb{S}_n)$ and prove that its Poincar\'e polynomial is asymptotically log-concave. Based on numerical computations with our algorithm, we further conjecture that the sequence $\{H^{2k}(\overline{\mathcal M}_{0,n})\}$ of $\mathbb{S}_n$-modules is equivariantly log-concave.