Talk:Longest element of a Coxeter group
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Completely wrong
editThis article mixes up two entirely different concepts: the longest element of a finite Weyl group W, usually denoted w0, and the Coxeter element. The two elements are almost never the same: the length of the former is the number of positive roots (n(n−1)/2 for the symmetric group Sn) and its order is always two; the length of the latter is the number of simple roots (n−1 for Sn), which is the same as the rank of the root system, and its order is the Coxeter number h. Unlike w0, the Coxeter element is defined for any, not necessarily finite, Weyl group. Why don't you use check a standard reference, such as Kac or Humphreys, instead of reinventing the wheel? By the way, there is no reason to create two separate articles, one for the Coxeter element and another for the Coxeter number. Arcfrk (talk) 23:38, 14 December 2008 (UTC)
- I have renamed the article and corrected the most egregious errors. A better long-term solution is to merge it as a section into Coxeter group. Arcfrk (talk) 00:19, 15 December 2008 (UTC)
- Thanks for the fixes, and apologies for the mistakes!
- As per “Summary style”, I believe this deserves its own article – Coxeter group is long enough as is, and even listing all the longest elements or properties of the longest element would bloat Coxeter group significantly.
- —Nils von Barth (nbarth) (talk) 10:08, 26 November 2009 (UTC)
Length of the longest element
editI wonder why the "Weyl group" condition is needed here. Reading Humphrey's book it seems to me that the length of the longest element is always equal to the number of positive roots, and that this has nothing to do with the crystallographic condition. The reason for this is that the length of an element w in a finite Coxeter group coincides with the number of positive roots that w maps to negative roots, and that this number is maximal exactly when all positive roots are mapped to negative ones. I therefore replaced "If the Coxeter group is a finite Weyl group then the length of w0 is the number of the positive roots." by "If the Coxeter group is finite then [...]". — Preceding unsigned comment added by 136.206.104.10 (talk) 17:06, 26 January 2017 (UTC)