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

Jinwon Choi; Young-Hoon Kiem; Donggun Lee

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arXiv · 2408.10728

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

Jinwon Choi·Young-Hoon Kiem·Donggun Lee

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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.

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BibTeX RIS

Jinwon Choi, Young-Hoon Kiem, Donggun Lee. 2024-08-20. Recursive algorithm and log-concavity of representations on the cohomology of $\overline{\mathcal M}_{0,n}$. https://arxiv.org/abs/2408.10728

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