The McKay correspondence ties the representation theory of a finite subgroup \(G\) of \(\operatorname{SL}(n,\mathbb{C})\) with resolution of singularities of the quotient orbifold \(\mathbb{C}^n/G\). More precisely, in dimensions 2 and 3
one can construct a crepant resolution: the \(G\)-Hilbert scheme, which is the fine moduli space of a certain collection of subschemes of \(\mathbb{C}^n\) called \(G\)-clusters. The 2-dimensional
correspondence is treated in full in the notes for my talk on Dynkin diagrams. It is such a concise correspondence due to the
strong classification of the finite subgroups of \(\operatorname{SL}(2,\mathbb{C})\). In four dimensions and above there is no general theory of crepant resolutions, and so three dimensions is a natural arena
for further study. The case of abelian subgroups of \(\operatorname{SL}(3,\mathbb{C})\) has an elegant treatment via toric geometry since \(G\)-\(\operatorname{Hilb}\) is itself toric in these cases, which stems from Nakamura (through combinatorial devices called
\(G\)-graphs that represent clever ways of writing a basis for the global sections of a \(G\)-cluster) and Craw-Reid. My work was to extend this to the non-abelian case. There is a whole
menagerie of finite non-abelian subgroups of \(\operatorname{SL}(3,\mathbb{C})\) and so my focus has been on trihedral groups: subgroups of the form \(A\rtimes T\) where \(A\) is a finite abelian subgroup of \(\operatorname{SL}(3,\mathbb{C})\) and \(T\) is
cyclic of order 3 generated by a permutation matrix. In this case I have an algorithm to compute \(G\)-\(\operatorname{Hilb}\) for trihedral groups via non-abelian analogues of \(G\)-graphs that are termed trihedral boats and 'non-abelian' representations of the McKay quiver.
Poster and
abstract for URSS research project supervised by Professor Miles Reid (FRS) at the University of Warwick during August-September 2013.
This is an example of the computation of clusters by trihedral boats. N.B. not this sort of boat.
For an introduction to some of the ideas underpinning the McKay correspondence and the setup for the trihedral case, see my master's thesis.
There is some unwritten material involving a description of general trihedral clusters and a correspondence between 'boundary' A-graphs and 'longboats'. For draft versions and some additional ramblings about operations on quiver representations,
see this or this.
You can find a more recent poster here.
See this chart for a diagram of Dynkin correspondences.
A chief reference for the McKay correspondence is Miles Reid's McKay page.
Research.