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.

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.