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A note on element centralizers in finite Coxeter groups. (English) Zbl 1245.20045
The main result that the authors obtain is the following theorem: Let \(W\) be a finite Coxeter group and let \(w\in W\). Let \(V\) be the smallest parabolic subgroup of \(W\) that contains \(w\). Then the following hold.
(i) The centralizer \(C_V(w)=C_W(w)\cap V\) is a normal subgroup of the centralizer \(C_W(w)\) with quotient \(C_W(w)/C_V(w)\) isomorphic to the normalizer quotient \(N_W(V)/V\).
(ii) The centralizer \(C_W(w)\) splits over \(C_V(w)\) with complement isomorphic to \(N_W(V)/V\) unless \(w\) lies in a non-compliant conjugacy class of elements of \(W\).
The authors also provide a short proof of a theorem of Solomon, and they discuss its relation to MacMahon’s master theorem.

MSC:
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20E07 Subgroup theorems; subgroup growth
20E45 Conjugacy classes for groups
Software:
CHEVIE; ZigZag
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