Hilbert series for twisted commutative algebras

Suppose that for each n >= 0 we have a representation $M_n$ of the symmetric group S_n. Such sequences arise in a wide variety of contexts, and often exhibit uniformity in some way. We prove a number of general results along these lines in this paper: our prototypical theorem states that if $M_n$ can be given a suitable module structure over a twisted commutative algebra then the sequence $M_n$ follows a predictable pattern. We phrase these results precisely in the language of Hilbert series (or Poincar'e series, or formal characters) of modules over tca’s.