  We develop a theory of levels for irreducible representations of symmetric
groups of degree $n$ analogous to the theory of levels for finite classical
groups. A key property of level is that the level of a character, provided it
is not too big compared to $n$, gives a good lower bound on its degree, and,
moreover, every character of low degree is either itself of low level or
becomes so after tensoring with the sign character. Furthermore, if $l_1$ and
$l_2$ satisfy a linear upper bound in $n$, then the maximal level of
composition factors of the tensor product of representations of levels $l_1$
and $l_2$ is $l_1+l_2$. To prove all of this in positive characteristic, we
develop the notion of rank, which is an analogue of the notion of rank of
cross-characteristic representations of finite classical groups. We show, using
modular branching rules and degenerate affine Hecke algebras, that the level
and the rank agree, as long as the level is not too large. We exploit
Schur-Weyl duality, modular Littlewood-Richardson coefficients and tilting
modules to prove a modular analogue of the Murnaghan-Littlewood theorem on
Kronecker products for symmetric groups. As an application, we obtain
representation growth results for both ordinary and modular representations of
symmetric and alternating groups analogous to those for finite groups of Lie
type.
