Special vs Essential

TL;DR

This paper extends the Special McKay correspondence by establishing a link between exceptional curves, divisors, and special representations of groups acting on complex three-space, providing explicit resolutions and a new correspondence.

Contribution

It introduces an explicit construction of small resolutions of G-Hilb(C^3) and links special and essential representations, extending known McKay correspondence results.

Findings

Established a correspondence between exceptional curves/divisors and special representations.

Provided an explicit construction of small resolutions of G-Hilb(C^3).

Extended the McKay correspondence and Reid's recipe.

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''.