  We present a lower bound on the Kronecker coefficients for tensor squares of
the symmetric group via the characters of~$S_n$, which we apply to obtain
various explicit estimates. Notably, we extend Sylvester's unimodality of
$q$-binomial coefficients $\binom{n}{k}_q$ as polynomials in~$q$ to derive
sharp bounds on the differences of their consecutive coefficients. We then
derive effective asymptotic lower bounds for a wider class of Kronecker
coefficients.
