Recursive algorithm and log-concavity of representations on the cohomology of M̄0,n

We provide a programmable recursive algorithm for the Sₙ-representations on the cohomology of the moduli spaces M̄0,n of n-pointed stable curves of genus 0. As an application, we find explicit inductive and asymptotic formulas for the invariant part H^*( M̄0,n/Sₙ) and prove that its Poincar\'e polynomial is asymptotically log-concave. Based on numerical computations with our algorithm, we further conjecture that the sequence ²ᵏ( M̄0,n)\ of Sₙ-modules is equivariantly log-concave.