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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.

MSC:
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)
Software:
GAP
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