2008/09/15 by Tathagata Basak, Basak, Tathagata
Mathematics · #20F05 #20F55 #20F65 #51F25 #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Random Matrices and Applications #Representation Theory (math.RT) #math.GR #math.RT #msc:20F05 #msc:20F55 #msc:20F65 #msc:51F25
paper · pdf · doi:10.48550/arxiv.0809.2427
27 pages, 4 figures. Major addition to the previous version. Section 4 is new. Organization of the paper modified. Stylistic changes. Small errors and typos corrected
openalex publication_date 2008/09/15 · arxiv created 2010/12/04 · arxiv updated 2010/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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[e2 πi/3]: there are only four such lattices, namely, the \cE-lattices whose real forms are A2, D4, E6 and E8. Next, we address the issue of characterizing the diagrams for unitary reflection groups, a question that was raised by Broué, 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 G33 and G34, new diagrams are obtained. For G34, our new diagram extends to an "affine diagram" with \ZZ/7\ZZ symmetry.