  The symmetric difference of the $q$-binomial coefficients
$F_{n,k}(q)={n+k\brack k}-q^{n}{n+k-2\brack k-2}$ was introduced by Reiner and
Stanton. They proved that $F_{n,k}(q)$ is symmetric and unimodal for $k \geq 2$
and $n$ even by using the representation theory for Lie algebras. Based on
Sylvester's proof of the unimodality of the Gaussian coefficients, as
conjectured by Cayley, we find an interpretation of the unimodality of
$F_{n,k}(q)$ in terms of semi-invariants. In the spirit of the strict
unimodality of the Gaussian coefficients due to Pak and Panova, we prove the
strict unimodality of the symmetric difference $G_{n,k,r}(q)={n+k\brack
k}-q^{nr/2}{n+k-r\brack k-r}$, except for the two terms at both ends, where
$n,r\geq8$, $k\geq r$ and at least one of $n$ and $r$ is even.
