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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
CHEVIE; ZigZag
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