 Unimodality of Gaussian binomial coefficients

For $0\le k\le n$, let $V$ be the $n$-dimensional irreducible sl2 Lie algebra representation. Its weights are $n-1,n-3,\dots ,1-n$. A wedge basis shows $\mathrm{ch}{\Lambda }^{k}V=q^{-k(n-k)}{G}_{n,k}(q^2)$, where $G_{n,k}(t)$ is the Gaussian binomial coefficient. By the Weyl complete reducibility theorem, this character is a nonnegative sum of irreducible strings $q^m+q^{m-2}+\cdots +q^{-m}$. All its weights have parity $k(n-k)$, so their coefficients increase toward zero and decrease afterward. Consequently the coefficients of $G_{n,k}(t)$ are symmetric and unimodal. The symmetric quantum binomial coefficient is this character, a Laurent polynomial; its coefficients are unimodal on the occupied parity. The full integer-exponent sequence need not be unimodal, as $q+q^{-1}$ demonstrates.