  We present several upper and lower bounds on the Kronecker coefficients of
the symmetric group. We prove $k$-stability of the Kronecker coefficients
generalizing the (usual) stability, and giving a new upper bound. We prove a
lower bound via the characters of $S_n$. We apply these and other results to
generalize Sylvester's unimodality of the $q$-binomial coefficients
$\binom{n}{k}_q$ as polynomials in $q$: we derive explicit sharp bounds on the
differences of their consecutive coefficients.
