Given a quiver Q=($Q_0,Q_1$) ($Q_0$ is the set of vertices and $Q_1$ is the set of arrows) and a dimension vector $\alpha$, the coordinate ring may be written as $\bigotimes_{a \in Q_1}k[Hom(k^{\alpha(ta)},k^{\alpha(ha)})]$ with ta, ha the tail and head of the arrow a. This ring is multigraded.
Kac's theorem on semi-invariants: Given an admissible vertex x(either a source or a sink), the semi-invariant ring on $Q,\alpha$ is isomorphic to the semi-invariant ring in $s_xQ,s_x\alpha$, where $s_x$ is the reflection around x, provided $s_x\alpha(x) \geq 0$.
My question is why the multidegree of a multihomogeneous semi-invariant in ($Q,\alpha$) remains unchanged in ($s_xQ,s_x\alpha$)?