  We initiate the systematic study of modular representations of symmetric
groups that arise via the braiding in (symmetric) tensor categories over fields
of positive characteristic. We determine what representations appear for
certain examples of tensor categories, develop general principles and
demonstrate how this question connects with the ongoing study of the structure
theory of tensor categories. We also formalise a theory of polynomial functors
as functors which act coherently on all tensor categories. We conclude that the
classification of such functors is a different way of posing the above question
of which representation of symmetric groups appear. Finally, we extend the
classical notion of strict polynomial functors from the category of (super)
vector spaces to arbitrary tensor categories, and show that this idea is also a
different packaging of the same information.
