Commuting graphs on Coxeter groups, Dynkin diagrams and finite subgroups of $SL(2,\mathbb{C})$
| Autor: | Hayat, Umar, de Celis, Álvaro Nolla, Ali, Fawad |
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| Rok vydání: | 2017 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | For a group $H$ and a non empty subset $\Gamma\subseteq H$, the commuting graph $G=\mathcal{C}(H,\Gamma)$ is the graph with $\Gamma$ as the node set and where any $x,y \in \Gamma$ are joined by an edge if $x$ and $y$ commute in $H$. We prove that any simple graph can be obtained as a commuting graph of a Coxeter group, solving the realizability problem in this setup. In particular we can recover every Dynkin diagram of ADE type as a commuting graph. Thanks to the relation between the ADE classification and finite subgroups of $\SL(2,\C)$, we are able to rephrase results from the {\em McKay correspondence} in terms of generators of the corresponding Coxeter groups. We finish the paper studying commuting graphs $\mathcal{C}(H,\Gamma)$ for every finite subgroup $H\subset\SL(2,\C)$ for different subsets $\Gamma\subseteq H$, and investigating metric properties of them when $\Gamma=H$. Comment: Re-estructured and large parts rewritten |
| Databáze: | arXiv |
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