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We study the $S_n$-equivariant log-concavity of the cohomology of flagvarieties, also known as the coinvariant ring of $S_n$. Using the theory ofrepresentation stability, we give computer-assisted proofs of the equivariantlog-concavity in low degrees and high degrees and conjecture that it holds forall degrees. Furthermore, we make a stronger unimodal conjecture which impliesthe equivariant log-concavity.

On the equivariant log-concavity for the cohomology of the flag varieties