= 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 $n-1,n-3,\ldots,1-n$. A wedge basis shows $\operatorname{ch}\Lambda^kV=q^{-k(n-k)}G_{n,k}(q^2)$, 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.
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