Sunday, 20 June 2010

rt.representation theory - Schemes of Representations of Groups

Let G be a group, say finitely presented as langlex1,ldots,xk|r1,ldots,rellrangle. Fix ngeq1 a natural number. Then there exists a scheme VG(n) contained in GL(n)k given by the relations. This scheme parameterizes n dimensional representations of G.



Now, I've known this scheme since I first started learning algebraic geometry (one of the first examples shown to me of an algebraic set was VS3(2)) but I've never found a good reference for this. So my first question is:



Is there a good reference for the geometry of schemes of representations?


Now, I have some much more specific questions. The main one being a point I'd been wondering about idly and tangentially since reading about some open problems related to the Calogero-Moser Integrable System:



Are there natural conditions on G that will guarantee that VG(n) be smooth? Reduced? Now, this is on the affine variety, I know that the projective closure will generally be singular, but in the case of VS3(2), I know that the affine variety defined above is actually smooth, of four irreducible components.



Finally, for any G and n, we have VG(n)subsetVG(n+1) (By taking the subscheme where the extra row and column are zeros, except on the diagonal, where it is 1). We can take the limit and get an ind-scheme, VG. What is the relationship between VG and the category Rep(G)? Can the latter be realized as a category of sheaves on the former? I know nothing here, and as I said, most of these questions are the result of idle speculation while reading about something else.



Edit: It occurs to me that as defined, VG(n) and VG may not be invariants of G, but really of the presentation. So two things to add: one, VG(n) is intended to really be the scheme Hom(G,GL(n)) (there's some issues I want to sweep under the rug with finitely generated infinite groups here, which is part of why I was thinking in presentations), and second, the situations that I'm thinking of are often the data of group with a presentation, so for that situation, VG(n) as defined should be good enough.

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