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- W4304699951 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 $Glhd widehat{G}$. 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 $Glhd widehat{G}$, 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." @default.
- W4304699951 created "2022-10-12" @default.
- W4304699951 creator A5068932564 @default.
- W4304699951 date "2022-10-08" @default.
- W4304699951 modified "2023-10-16" @default.
- W4304699951 title "Real McKay Correspondence: KR-Theory of Graded Kleinian Groups" @default.
- W4304699951 doi "https://doi.org/10.48550/arxiv.2210.03924" @default.
- W4304699951 hasPublicationYear "2022" @default.
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