Powers of Coxeter elements in infinite groups are reduced

By David E. Speyer. 2007

63 citations recorded by the source corpus.

Let W be an infinite irreducible Coxeter group with (s 1 ,..., s n ) the simple generators. We give a short proof that the word s 1 s 2 ... s n s 1 s 2 ··· s n ···s 1 s 2 ...s n is reduced for any number of repetitions of s 1 s 2 ··· s n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley 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 1 s 2 ···s n s 1 s 2 ··· s n ··· by interchanging commuting elements and taking a prefix.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/3cb986d4565a2b4ab5a56bf6c1adefa17d989aed

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In Discrete Mathematics and Combinatorics

  1. S. Nie
  2. Xuhua He
  3. Michael Rapoport

In Geometry and Topology

  1. S. Nie
  2. Xuhua He
  3. Michael Rapoport

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