I would like to know, whether the following quotient construction has been considered, or whether it makes sense:
One way to think of toric varieties is as a quotient of $\mathbb{C}^n$ (minus exceptional set) by a product of $\mathbb{C}^*$'s, giving different projective weights to the $n$ complex coordinates. Choosing the weights amounts to setting the representation of $\mathbb{C}^*$ under which a homogeneous coordinate transforms. In the quiver GIT constructions, one also considers quotients by products of $GL$ groups. However, in that case the representations are always the trivial, fundamental or anti-fundamental representations, since quiver arrows don't allow for more.
Are there constructions where one uses more general representations of $GL$ to create such quotients?