zbMATH — the first resource for mathematics

On reflection subgroups of finite Coxeter groups. (English) Zbl 1282.20040
Let \((W,S)\) be a finite Coxeter system with distinguished set of generators \(S\). The authors give a complete classification of all reflection subgroups of \(W\) up to conjugacy.
Let \(\mathfrak R\) be the set of conjugacy classes of reflection subgroups of \(W\) and let \(\mathfrak C\) be the set of conjugacy classes of elements of \(W\). Let \(\gamma\colon\mathfrak R\to\mathfrak C\) be the map defined by \(\gamma([R])=[c]\), which associates to the conjugacy class \([R]\) of a reflection subgroup \(R\) of \(W\) the conjugacy class \([c]\) in \(W\) of a Coxeter element \(c\) in \(R\).
The authors compute the image of \(\gamma\) explicitly for each type of irreducible Coxeter group.

20F55 Reflection and Coxeter groups (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
20E45 Conjugacy classes for groups
05E15 Combinatorial aspects of groups and algebras (MSC2010)
Full Text: DOI
[1] DOI: 10.1017/S0305004100052610 · Zbl 0364.22007 · doi:10.1017/S0305004100052610
[2] DOI: 10.1023/A:1022481230120 · Zbl 0798.20031 · doi:10.1023/A:1022481230120
[3] DOI: 10.1007/BF02565599 · Zbl 0034.30701 · doi:10.1007/BF02565599
[4] Bourbaki N., 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 (1968) · Zbl 0186.33001
[5] Carter R. W., Compositio Math. 25 pp 1– (1972)
[6] DOI: 10.1016/0021-8693(72)90062-2 · Zbl 0239.20053 · doi:10.1016/0021-8693(72)90062-2
[7] DOI: 10.1515/jgt.2009.061 · Zbl 1192.20023 · doi:10.1515/jgt.2009.061
[8] Dyer , M. J. , Lehrer , G. I. ( 2011 ). Reflection subgroups of finite and affine Weyl groups . · Zbl 1243.20051
[9] DOI: 10.1007/BF01190329 · Zbl 0847.20006 · doi:10.1007/BF01190329
[10] Geck , M. , Pfeiffer , G. ( 2000 ). Characters of finite Coxeter groups and Iwahori-Hecke algebras.London Mathematical Society Monographs. New Series.Vol. 21. New York: The Clarendon Press Oxford University Press . · Zbl 0996.20004
[11] DOI: 10.1090/pspum/040.2/713255 · doi:10.1090/pspum/040.2/713255
[12] GAP – Groups, Algorithms, and Programming (1995)
[13] DOI: 10.1016/0021-8693(76)90182-4 · Zbl 0355.20007 · doi:10.1016/0021-8693(76)90182-4
[14] DOI: 10.1090/S0002-9947-1964-0167535-3 · doi:10.1090/S0002-9947-1964-0167535-3
[15] DOI: 10.4213/sm1422 · doi:10.4213/sm1422
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.