# Representations on the cohomology of smooth projective hypersurfaces with symmetries

## TL;DR

This paper investigates how the automorphism group acts on the primitive cohomology of smooth projective hypersurfaces, providing character formulas and implications for symmetric hypersurfaces invariant under permutation groups.

## Contribution

It introduces a method to compute the automorphism group action on cohomology using Lefschetz-Riemann-Roch and explores symmetric hypersurfaces stable under permutation group actions.

## Key findings

- Character formulas for automorphism group actions on cohomology
- Existence results for symmetric hypersurfaces with permutation symmetry
- Application of Lefschetz-Riemann-Roch to geometric representation theory

## Abstract

This paper is concerned with the primitive cohomology of a smooth projective hypersurface considered as a linear representation for its automorphism group. Using the Lefschetz-Riemann-Roch formula, the character of this representation is described on each piece of the Hodge decomposition. A consequence about the existence of smooth symmetric hypersurfaces that are stable under the standard irreducible permutation representation of the symmetric group on homogeneous coordinates is drawn.