Moduli of constellations/representations

This is more an aside to the previous post than an independent account; the objective was to fill in some of the details of the construction and manipulation of the moduli spaces involved. These spaces parameterise G-constellations – equivariant coherent sheaves on \mathbb{A}^n whose global sections are isomorphic to the regular representation as G-modules – or, equivalently, representations of the McKay quiver of G with a natural choice of dimension vector and relations.

It appears to me that there are two main approaches to constructing and parameterising G-constellations. I call them ‘combinatorial’ and ‘algebraic’, although there are clearly combinatorial elements in the algebraic version, and vice versa.

Combinatorial: start with the coordinate ring k[\mathbb{A}^n] on which G acts and find quotients of it that are isomorphic to k[G]. This approach was initiated by Ito, Nakamura, and Reid most notably via the technology of G-graphs. This is good for a local description of G\text{-Hilb} among other things.

Algebraic: start with the regular representation k[G] and find k[\mathbb{A}^n]-actions on it that are compatible with the action of G. This is primarily the approach taken by people like Craw and Ishii especially when using representations of quivers, which very visually document the action of k[\mathbb{A}^n].

The nonabelian case presents challenges for whichever approach is taken (usually a mixture of the two). For example, more complicated relations and the sheer number of variables defining G-constellations are computational difficulties that one meets almost immediately.

These rough notes also take the opportunity to discuss stability of equivariant sheaves and, again equivalently, quiver representations and the role it plays in constructing and studying the moduli spaces at work in the McKay correspondence.


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