The action of S_n on the cohomology of M_{0,n}(R)

TL;DR

This paper determines how the symmetric group acts on the cohomology of the real moduli space of genus 0 curves with n marked points, providing explicit formulas for the character of this action.

Contribution

It offers explicit formulas for the graded character of the symmetric group action on the cohomology of _{0,n}(R), extending previous structural results.

Findings

Derived a plethystic formula for the cycle index of the action.

Provided an explicit product formula for the character on cycle types.

Connected the group action to the cohomological structure of the moduli space.

Abstract

In recent work (math/0507514) by Etingof, Henriques, Kamnitzer, and the author, a presentation and explicit basis was given for the rational cohomology of the real locus $\bar{M_{0,n}}(\RR)$ of the moduli space of stable genus 0 curves with $n$ marked points. We determine the graded character of the action of $S_n$ on this space (induced by permutations of the marked points), both in the form of a plethystic formula for the cycle index, and as an explicit product formula for the value of the character on a given cycle type.