Powers of Coxeter elements in infinite groups are reduced. (English) Zbl 1187.20053
Summary: Let \(W\) be an infinite irreducible Coxeter group with \((s_1,\dots,s_n)\) the simple generators. We give a short proof that the word \(s_1s_2\cdots s_ns_1s_2\cdots s_n\cdots s_1s_2\cdots s_n\) is reduced for any number of repetitions of \(s_1s_2\cdots s_n\). This result was proved for simply laced, crystallographic groups by M. Kleiner and A. Pelley [Int. Math. Res. Not. 2007, No. 4, Article ID rnm013 (2007; Zbl 1135.16014)] using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for \(w_0\) which is obtained from the semi-infinite word \(s_1s_2\cdots s_ns_1s_2\cdots s_n\cdots\) by interchanging commuting elements and taking a prefix.
MSC:
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 20F05 | Generators, relations, and presentations of groups |
| 17B22 | Root systems |
Keywords:
infinite irreducible Coxeter groups; simple generators; reduced words; root systems; finite Coxeter groupsCitations:
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