Real McKay Correspondence: KR-Theory of Graded Kleinian Groups

TL;DR

This paper investigates the $KR$-theory of graded Kleinian groups, extending the McKay correspondence to include involutions and real structures, with detailed calculations of Frobenius-Schur indicators and decorated McKay graphs.

Contribution

It introduces a conjectural extension of the McKay correspondence using $KR$-theory for $C_2$-graded groups, incorporating involutions and real structures.

Findings

Computed Frobenius-Schur indicators for index 2 subgroup containments.

Produced decorated McKay graphs for each case.

Conjectured a $KR$-theoretic analogue of the McKay correspondence.

Abstract

This project considers the finite symmetry subgroups of the orthogonal group $\mathrm{O}(3) \subset \mathrm{GL}(3,\mathbb{R})$ and the index $2$ containments $G\lhd \widehat{G}$. The special orthogonal group $\mathrm{SO}(3) \subset \mathrm{SL}(3,\mathbb{R})$ admits a double cover from the spinor group $\mathrm{Spin}(3) \cong \mathrm{SU}(2) \subset \mathrm{SL}(2,\mathbb{C})$, and lifting our subgroups up preserves the network of containments. Those subgroups not contained in $\mathrm{SO}(3) \subset \mathrm{O}(3) $ are lifted to the pinor groups $\mathrm{Pin}_{\pm}(3)$ of which there are two choices. For the index $2$ containments $G\lhd \widehat{G}$, we calculate the Real and complex Frobenius-Schur indicators, and apply Dyson's classification of antilinear block structures to produce decorated McKay graphs for each case. We then explore $KR$-theory as introduced by Atiyah in 1966, which is a variant of topological $K$-theory for working with topological spaces equipped with an involution. The GIT quotient spaces $\mathbb{C}^2 // G$, can be equipped by an involution via the action of $\widehat{G} / G$. In 1983, Gonzalez-Sprinberg and Verdier showed how one can view the McKay correspondence as an isomorphism between the $G$-equivariant $K$-theory $K_G(\mathbb{C}^2)$ and the $K$-theory of the minimal resolution of the singularity $\widetilde{\mathbb{C}^2 // G}$. We use this to conjecture an analogous a form of the McKay correspondence for $C_2$-graded groups and $KR$-theory.