论文标题
真正的McKay通信：分级Kleinian群体的KR理论
Real McKay Correspondence: KR-Theory of Graded Kleinian Groups
论文作者
论文摘要
该项目考虑了正交组$ \ mathrm {o}（3）（3）\ subset \ mathrm {gl}（3，\ mathbb {r}）$和索引$ 2 $ continments $ g \ g \ lhd \ lhd \ wideHat {g} $。特殊的正交组$ \ mathrm {so}（3）\ subset \ mathrm {sl}（3，\ mathbb {r}）$允许旋转组$ \ mathrm {spin}（spin}（3）（3）\ cong \ mathrm {surm {su}（2）（2）\ suex（2）（2）\ suex（2）\ suex（2）） \ Mathrm {Sl}（2，\ Mathbb {C}）$，并提起我们的子组保留了封存网络。这些子组未包含在$ \ mathrm {so}（3）（3）\ subset \ mathrm {o}（3）$中，将其提升到Pinor组$ \ Mathrm {Pin} _ {\ pm}（3）$，其中有两种选择。对于索引$ 2 $ continments $ g \ lhd \ wideHat {g} $，我们计算了真实且复杂的Frobenius-Schur指示器，并应用Dyson对Antinear Block结构的分类来为每种情况生成装饰的McKay图形。然后，我们探索了Atiyah在1966年介绍的$ KR $理论，这是拓扑$ K $的一种变体，用于与配备有相关性的拓扑空间合作。 git商空间$ \ mathbb {c}^2 // g $，可以通过$ \ wideHat {g} / g $的操作来配备互动。 1983年，冈萨雷斯·史普林伯格（Gonzalez-Sprinberg）和凡尔迪尔（Verdier）展示了人们如何将麦凯对应关系视为$ g $ - equivariant $ k $ -k $ - 理论$ k_g（\ mathbb {c}^2）$与$ k $ - $ k $ - singularity $ \ \ widetildede的最低分辨率的理论} G} $。我们用它来猜测$ C_2 $分组的McKay通信形式和$ KR $ - 理论。
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.
