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The first question might be too much in general.

The cases I'd like to understand in practice are quotients (as algebraic varieties) of GL(n,C) (or SL(n,C) if you prefer) by finite subgroups. Is there anything I can say about the quotient (set of cosets)?

The babiest case would be the standard rep of the dihedral/symmetric group $S_3$. What is $GL(2,C)/S_3$? Or $SL(2,C)/S_3$ if it makes things easier?

Essentially what I'm trying to do is embed a finite group in a special group (in the technical sense) and understanding what the quotient is.

Any ideas or references?

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  • $\begingroup$ I find your question too vague. For example, there are many embeddings of the same finite cyclic group in $SO_4$. So the quotients are all different, and there's different "levels" of differentness depending on what equivalence relation you use on the quotient spaces. Most of them are quite interesting and involved but if you don't state your preference it's not clear where to go. $\endgroup$
    – Ryan Budney
    Commented Aug 1, 2011 at 0:07
  • $\begingroup$ Yes my question is vague. But can we say anything about the 'easy' case I mention? The one with the dihedral group and its standard representation? Is the quotient isomorphic to something familiar? $\endgroup$
    – Yosemite Sam
    Commented Aug 1, 2011 at 0:18
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    $\begingroup$ I'm afraid my answer is slightly tautological, but $GL_2(\mathbb{C})/S_3$ is just the space of unordered triples of non-zero vectors $\{v_1,v_2,v_3\}$ from $\mathbb{C}^2$, such that $v_1+v_2+v_3=0$ (just like $GL_2(\mathbb{C})$ is the set of linear bases in $\mathbb{C}^2$). $\endgroup$
    – Alain Valette
    Commented Aug 1, 2011 at 3:13
  • $\begingroup$ Here is another trivial remark: if $X$ is an affine variety over the complex numbers and $G$ is a finite group acting on $X$, then $X/G=\mathbb{C}[X]^G$, which is again an affine variety. Of course, one can say more for some particular $X$ and $G$. $\endgroup$
    – algori
    Commented Aug 1, 2011 at 3:49
  • $\begingroup$ Small correction about my previous comment. It's not enough for the 3 vectors to be non-zero, you need them to span $\mathbb{C}^2$. More geometrically, $GL_2(\mathbb{C})/S_3$ is the space of triangles in $\mathbb{C}^2$, spanning $\mathbb{C}^2$ and barycentered at the origin. $\endgroup$
    – Alain Valette
    Commented Aug 1, 2011 at 4:05

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