Bounds on certain classes of 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 Sn, which we apply to obtain various explicit estimates. Notably, we extend Sylvester's unimodality of q-binomial coefficients (nk)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.