Bounds on Kronecker and $q$-binomial coefficients

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 ${\left(\genfrac{}{}{0px}{}{n}{k}\right)}_q$ 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.