  There is growing experimental interest in coupling cavity photons to the
cyclotron resonance excitations of electron liquids in high-mobility
semiconductor quantum wells or graphene sheets. These media offer unique
platforms to carry out fundamental studies of exciton-polariton condensation
and cavity quantum electrodynamics in a regime in which electron-electron
interactions are expected to play a pivotal role. Focusing on graphene, we
present a theoretical study of the impact of electron-electron interactions on
a quantum Hall polariton fluid, that is a fluid of magneto-excitons resonantly
coupled to cavity photons. We show that electron-electron interactions are
responsible for an instability of graphene integer quantum Hall polariton
fluids towards a modulated phase. We demonstrate that this phase can be
detected by measuring the collective excitation spectra, which soften at a
characteristic wave vector of the order of the inverse magnetic length.
