# Special Year Seminar II

## Representations on the Cohomology of the Moduli Space of Pointed Rational Curves

The moduli space of pointed rational curves has a natural action of the symmetric group permuting the marked points. In this talk, we will present a combinatorial formula for the induced representation on the cohomology of the moduli space, along with a recursive algorithm derived from this formula. These results arise from wall crossing phenomena among birational models of the moduli space, governed by Hassett’s theory of weighted stable curves and Choi-Kiem’s theory of delta-stability of quasimaps.

These results enable us to study positivity and log-concavity of the representation. Specifically, we will provide partial affirmative answers to the question of whether the representation is permutation representation in each degree. We will also present explicit inductive and asymptotic formulas for the multiplicity of the trivial representation and confirm its asymptotic log-concavity. If time permits, we will also discuss the multiplicities of other irreducible representations as well. This talk is based on joint works with Jinwon Choi and Young-Hoon Kiem.