The Cautis-Logvinenko conjecture

For a finite subgroup $G\subset \mathrm{SL}(3,\mathbb{C})$, the Cautis–Logvinenko conjecture states that for each nontrivial irreducible representation $\rho$ of $G$, the image of the sheaf ${\mathcal{O}}_0\otimes \rho$ under the derived equivalence of Bridgeland–King–Reid is a pure sheaf on the $G$-Hilbert scheme. We prove this when the McKay quiver of $G$ contains no loops; this includes many dihedral and trihedral subgroups of $\mathrm{SL}(3,\mathbb{C})$, as well as six of the eight sporadic finite subgroups. In doing so, we compute the relevant sheaf explicitly whenever its support is of dimension one. Our main result implies that a matrix defining the Gale dual of the linearisation map is sign-coherent, thereby allowing us to read off the support and cohomological degree of the pure sheaves directly from the matrix.