Longest element of Weyl groups. - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T21:45:43Z http://mathoverflow.net/feeds/question/54926 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54926/longest-element-of-weyl-groups Longest element of Weyl groups. unknown (google) 2011-02-09T20:53:45Z 2012-03-23T23:43:11Z <p>What is a reduced expression of the longest element of each type of Weyl group. For type $A_n$ it is just $s_n(s_ns_{n-1})...(s_n...s_1)$. I know for type $B_n,C_n,E_7,E_8$,$G_2$ and $D_n$ (n even) it is just $-id$, although I don't have an explicit reduced expression for them. For type $D_n$ (n odd) and type $E_6$ I don't know what are the longest elements. Any reference where it is written explicitely ? </p> http://mathoverflow.net/questions/54926/longest-element-of-weyl-groups/54934#54934 Answer by Petra Schwer for Longest element of Weyl groups. Petra Schwer 2011-02-09T21:57:27Z 2011-02-09T21:57:27Z <p>A good reference to try is Bourbaki "Lie groups and Lie Algebras, Chapters 4-6" Look at the plates at the end of the book, which contain all kinds of useful information about each one of the types. </p> http://mathoverflow.net/questions/54926/longest-element-of-weyl-groups/54944#54944 Answer by Steven Sam for Longest element of Weyl groups. Steven Sam 2011-02-09T22:47:03Z 2011-02-09T22:47:03Z <p>You can get some reduced expressions using LiE: <a href="http://www-math.univ-poitiers.fr/~maavl/LiE/" rel="nofollow">http://www-math.univ-poitiers.fr/~maavl/LiE/</a></p> <p>via the command long_word(Xn) where Xn is the Dynkin diagram.</p> http://mathoverflow.net/questions/54926/longest-element-of-weyl-groups/54947#54947 Answer by Jim Humphreys for Longest element of Weyl groups. Jim Humphreys 2011-02-09T22:54:53Z 2012-03-23T23:43:11Z <p>EDIT: This is a belated attempt (motivated by a question from Yongjun Xu) to answer the question more precisely than I did at first, with more emphasis on careful choice of a Coxeter element when the Coxeter number <code>$h$</code> is <em>even</em> (as happens for all irreducible Weyl groups except those of type <code>$A_\ell$</code> with <code>$\ell$</code> even). As Allen points out, for even type <code>$A$</code> the algorithm needs to be modified somewhat. </p> <p>I think the best conceptual approach (leaving even type <code>$A$</code> aside) is based on the Coxeter element, treated in detail in Bourbaki's Chapter V and later in Chapter 3 of my 1990 book <em>Reflection Groups and Coxeter Groups</em>. My exercise 2 at the end of 3.19 deals with the longest element explicitly, but requires a special choice of the Coxeter element <code>$w$</code> as in the treatment of the Coxeter plane in 3.17: here you start by dividing the simple reflections into two sets, each consisting of mutually orthogonal reflections. For instance, with the usual numbering in type <code>$A_3$</code> you can take <code>$w = s_1 s_3 s_2$</code> but not <code>$w =s_1 s_2 s_3$</code> (whose square is non-reduced). Then you automatically get a reduced expression <code>$w_\circ = w^{h/2}$</code> for the longest element. </p> <p>Bourbaki discusses all of this in section 6.2: see Proposition 2, where it is understood that <code>$W$</code> is irreducible and the Coxeter element (denoted <code>$c$</code>) has the special format fixed earlier in the section. (The wording of my exercise was corrected a long time ago in the list of revisions which I keep on my homepage, but my original answer here overlooked that correction. After the 1992 reprinting of my book, publishers began using print-on-demand technology involving photocopies, so no further corrections can be made.) </p> <p>As in Bourbaki's Chapter V, the treatment actually involves arbitrary irreducible finite reflection groups, not just Weyl groups. Besides even rank <code>$A_\ell$</code>, the odd rank dihedral groups have to be omitted because their Coxeter numbers are odd. </p> http://mathoverflow.net/questions/54926/longest-element-of-weyl-groups/54977#54977 Answer by Allen Knutson for Longest element of Weyl groups. Allen Knutson 2011-02-10T02:21:32Z 2011-02-10T13:30:55Z <p>2-color your Dynkin diagram, black and white. Let $w$ be the product of the white simple reflections, $b$ the product of the black. Note that $w$ and $b$ are well-defined, as the reflections you're multiplying to make them, commute. You'll have to pick the order if you want an actual word, in what follows.</p> <p>If $G$ is not $A_{even}$: the affinization of the diagram is also 2-colorable, so you can choose the affine vertex to be white. Let $\chi = w b$, a Coxeter element. The Coxeter number $h$ is even, and $\chi^{h/2} = w_0$. So you get a reduced word $wbwbwb\ldots wb$, where the total number of letters is $h$ (and each letter is a product of commuting reflections).</p> <p>If $G$ is, unfortunately, $A_{even}$: you have to pick $w$ vs. $b$, and the diagram automorphism shows that the choice is unavoidable. The Coxeter number is odd. But you still get a reduced word, $wbwb\ldots bw$, again with $h$ letters.</p>