  The composition in terms of nuclear species of the primary cosmic ray flux is
largely uncertain in the knee region and above, where only indirect
measurements are available. The predicted fluxes of high-energy leptons from
cosmic ray air showers are influenced by this uncertainty. Different models
have been proposed. Similarly, these uncertainties affect the measurement of
lepton fluxes in very large volume neutrino telescopes. Uncertainties in the
cosmic ray interaction processes, mainly deriving from the limited amount of
experimental data covering the particle physics at play, could also produce
similar differences in the observable lepton fluxes and are affected as well by
large uncertainties. In this paper we analyse how considering different models
for the primary cosmic ray composition affects the expected rates in the
current generation of very large volume neutrino telescopes (ANTARES and
IceCube). We observe that, a certain degree of discrimination between
composition fits can be already achieved with the current IceCube data sample,
even though in a model-dependent way. The expected improvements in the energy
reconstruction achievable with the next generation neutrino telescopes is be
expected to make these instruments more sensitive to the differences between
models.
