Special vs Essential

Abstract

We show a correspondence between the compact exceptional curves and divisors on $G-{\rm Hilb}(\mathbf{C}^3)$ and some non-trivial irreducible representations of $G \subset GL(n,C)$ which are special (or essential). Moreover, we provide an explicit construction of the small resolution of $G-{\rm Hilb}(\mathbf{C}^3)$ and, using this resolution, we construct a correspondence between special and essential representations. These results are an extension of ``Special McKay correspondence'' and ``Reid's recipe''.