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- W4321153939 abstract "We study Coxeter diagrams of some unitary reflection groups. Using solely the combinatorics of diagrams, we give a new proof of the classification of root lattices defined over $cE = ZZ[e^{2 pi i/3}]$: there are only four such lattices, namely, the $cE$-lattices whose real forms are $A_2$, $D_4$, $E_6$ and $E_8$. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Brou'{e}, Malle and Rouquier. To this end, we describe an algorithm which, given a unitary reflection group $G$, picks out a set of complex reflections. The algorithm is based on an analogy with Weyl groups. If $G$ is a Weyl group, the algorithm immediately yields a set of simple roots. Experimentally we observe that if $G$ is primitive and $G$ has a set of roots whose $ZZ$--span is a discrete subset of the ambient vector space, then the algorithm selects a minimal generating set for $G$. The group $G$ has a presentation on these generators such that if we forget that the generators have finite order then we get a (Coxeter-like) presentation of the corresponding braid group. For some groups, such as $G_{33}$ and $G_{34}$, new diagrams are obtained. For $G_{34}$, our new diagram extends to an affine diagram with $ZZ/7ZZ$ symmetry." @default.
- W4321153939 created "2023-02-18" @default.
- W4321153939 creator A5085174191 @default.
- W4321153939 date "2008-09-14" @default.
- W4321153939 modified "2023-09-29" @default.
- W4321153939 title "On Coxeter Diagrams of complex reflection groups" @default.
- W4321153939 doi "https://doi.org/10.48550/arxiv.0809.2427" @default.
- W4321153939 hasPublicationYear "2008" @default.
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