Automorphisms of Coxeter groups of rank three

Franzsen, W.; Howlett, R.

doi:10.1090/S0002-9939-01-05878-6

Automorphisms of Coxeter groups of rank three
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by W. N. Franzsen and R. B. Howlett PDF

Proc. Amer. Math. Soc. 129 (2001), 2607-2616 Request permission

Abstract:

If $W$ is an infinite rank $3$ Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of $W$ is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed $\operatorname {Aut}(W)$ is the semi-direct product of $\operatorname {Inn}(W)$ and the group of graph automorphisms.

References

Brigitte Brink, The set of dominance-minimal roots, J. Algebra 206 (1998), no. 2, 371–412. MR 1637139, DOI 10.1006/jabr.1997.7418
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
Vinay V. Deodhar, On the root system of a Coxeter group, Comm. Algebra 10 (1982), no. 6, 611–630. MR 647210, DOI 10.1080/00927878208822738
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646