Minimal length elements of finite Coxeter groups. (English) Zbl 1272.20042

He, Xuhua; Nie, Sian

Duke Math. J. 161, No. 15, 2945-2967 (2012).

Given a group and an automorphism let us consider the twisted conjugacy classes (also known as Reidemeister classes). With respect to a set of generators, one can look at the elements of one twisted class which have minimal length. One would like to study certain properties of those elements of minimal length in some concrete cases. Those properties have been studied by M. Geck et al., [J. Algebra 229, No. 2, 570-600 (2000; Zbl 1042.20026); Adv. Math. 102, No. 1, 79-94 (1993; Zbl 0816.20034)], and X. He, [Adv. Math. 215, No. 2, 469-503 (2007; Zbl 1149.20035)] when the group \(G\) is a finite Coxeter group and the automorphism permutes the reflections of the finite Coxeter group. The properties, refered to as “remarkable properties”, roughly speaking consist in being able to move from any element of the twisted conjugacy class to an element of minimal length using a finite number of times the twisted relation with respect to elements of the group which are reflections in such way that the length of the word in each step does not increase.
The main result of the paper is: Theorem. Let \((W,S)\) be a Coxeter group, and let \(\delta\colon W\to W\) be an automorphism sending simple reflections to simple reflections. Let \(\mathcal O\) be a \(\delta\)-twisted conjugacy class in \(W\), and let \(\mathcal O_{\min}\) be the set of minimal length elements in \(\mathcal O\). Then
(1) for each \(w\in\mathcal O\), there exists \(w'\in\mathcal O_{\min}\) such that \(w\to_\delta w'\);
(2) let \(w,w'\in\mathcal O_{\min}\); then \(w\sim w'\).
The above theorem is proved in two papers refered to above via case-by-case analysis. The main goal of the paper under review is to give a case-free proof of this result. As important tool, the authors give a geometric interpretation of the twisted classes using an embbeding of \(\langle\delta\rangle\ltimes W\) on \(\mathrm{GL}(V)\) where \(V\) is a vector space. After they prove the main result in section 3, they then specialize to the so called elliptic conjugacy classes and obtain even stronger results than the main one. At the end they look to the braid monoid associated with \((W,S)\) and they study good elements of the braid monoid in the same spirit as they have done with the twisted classes in section 3. In the introduction one finds several comments which indicate other advantages and consequences of the approach used.

Reviewer: Daciberg Lima Gonçalves (São Paulo)

MSC:

20F55	Reflection and Coxeter groups (group-theoretic aspects)
20E45	Conjugacy classes for groups
51F15	Reflection groups, reflection geometries
55M20	Fixed points and coincidences in algebraic topology

Keywords:

Coxeter groups; twisted conjugacy classes; reflections; minimal length elements; elliptic conjugacy classes; braid monoids

Citations:

Zbl 1042.20026; Zbl 0816.20034; Zbl 1149.20035

References:

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