3
$\begingroup$

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?

Improve this question
$\endgroup$
2
  • $\begingroup$ Sure: for any complex reductive group G and any linear representation V of G, you can consider the GIT quotient G//V (which in general can be twisted by a choice of character of G). Toric varieties are obtained for G a torus (product of $\mathbb{C}^{*}$) and quiver varieties for G a product of linear groups and V made of fundamentals and anti-fundamentals. $\endgroup$
    – user25309
    Jan 23, 2017 at 11:11
  • $\begingroup$ Has this been used explicitly for the resolved du Val singularities? I have seen those described via quiver GIT, but then those spaces require preprojective algebras of quadratic relations. I always suspected this was due to the limitation in representations used in quivers. For instance, the A series can be treated torically, without resort to relations if one allows for more general reps. Could the D and E series be treated as "appropriate" (meaning not using the (anti-)fundamental)) quotients of GL groups? $\endgroup$
    – Andres Collinucci
    Jan 23, 2017 at 11:37

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.