The magic square of reflections and rotations. (English) Zbl 07784571

In the paper under review, the authors consider the following four classes of groups (up to conjugation): (a) finite subgroupsof \(\mathrm{SL}(2,\mathbb{C})\) of even order, (b) finite reflection subgroups of \(\mathrm{GL}(2, \mathbb{C})\) containing \(-1\), (c) finite subgroups of \(\mathrm{SL}(3,\mathbb{R})\), (d) finite reflection subgroups of \(\mathrm{GL}(3,\mathbb{R})\). Among the sets just defined there are some well-known bijections. In particular, these bijections have been studied in the case from (d) to (a) by C. Jordan [Borchardt J. LXXXIV, 89–215 (1877; JFM 09.0096.01)] and by F. Klein [Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade. Leipzig. Teubner (1884; JFM 16.0061.01)], from (c) to (d) by H. M. S. Coxeter [Ann. Math. (2) 35, 588-621 (1934; JFM 60.0898.02)], from (b) to (c) by D. Bessis et al. [Math. Ann. 323, No. 3, 405–436 (2002; Zbl 1053.20037)] and from (a) to (b) by G. C. Shephard and J. A. Todd [Can. J. Math. 6, 274–304 (1954)] and by G. Gonzalez-Sprinberg and J. L. Verdier [Ann. Sci. Éc. Norm. Supér. (4) 16, 409–449 (1983; Zbl 0538.14033)]. The title of the paper refers to the square diagram (a), (b), (c), (d). The authors survey how Coxeter’s work implies a bijection between complex reflection groups of rank two and real reflection groups in \(\mathrm{O}(3)\). They also consider this “magic” square of reflections and rotations in the framework of Clifford algebras. In particular, they give an interpretation using (s)pin groups and explore these groups in small dimensions.


20F55 Reflection and Coxeter groups (group-theoretic aspects)
11E88 Quadratic spaces; Clifford algebras
14E16 McKay correspondence
15A66 Clifford algebras, spinors
51F15 Reflection groups, reflection geometries
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