Is there any relation between length function of a coxeter group with respect to two different simple systems as two simple systems are weyl conjugates of one another?
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2$\begingroup$ I'm a little unclear on what you are asking (because I do not know what a "simple system" is), but here is a general fact. Given a group $W$ with two finite generating sets $S_1,S_2$ and their associated length functions $L_1$ and $L_2$, if $K$ is the maximum of $L_1(s), L_2(s)$ over all $s \in S_1 \cup S_2$ then $L_1(w)/K \le L_2(w) \le K L_1(w)$ for all $w \in W$. $\endgroup$– Lee MosherCommented Dec 3, 2012 at 14:29
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$\begingroup$ As David Speyer notes, this can't be so, for fairly obvious reasons. But one's feeling that there is some "metric object" that is independent of generators is correct, namely, the "gallery-distance" on the associated Coxeter complex. The word-length is the gallery-distance from the distinguished chamber (reflections through whose sides are the given generators), to its image under the word. $\endgroup$– paul garrettCommented Dec 3, 2012 at 15:38
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2 Answers
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To supplement what David Speyer and Paul Garrett have written, I'd emphasize that the question itself is faulty: the terminology involved in "two simple systems are weyl conjugates of one another" doesn't make sense here. Whenever one speaks of a Coxeter group one is implicitly referring to a pair $(W,S)$ consisting of a group $W$ and a specified set $S$ of generators, along with certain explicit relations among those generators. Only when you fix this data do you get a definite length function (relative to $S$) for $W$.
The tag "lie-algebras" already indicates some fuzziness here: there is no Lie algebra and no Weyl group. Appropriate tags include "gr.group-theory" and "coxeter-groups" As David's example indicates, even for simple Lie algebras and their Weyl groups, the Weyl group as a Coxeter group may admit multiple unrelated presentations. And even if one requires $S$ to be a set of simple reflections in the sense of Lie theory, any root can be a simple root and have length 1 relative to some choice of $S$. So I'm not sure what the question is really about(?)
answered Dec 3, 2012 at 19:51
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1$\begingroup$ @Jim: Things are actually even more interesting, since one can have two Coxeter systems $(W_1,S_1), (W_2,S_2)$, where $W_1$ is isomorphic to $W_2$ as an abstract group, but that there is no isomorphism of Coxeter systems. Thus, the entire premise of $S_1$ being conjugate to $S_2$ is false. On the other hand, in many cases, Coxeter groups are "rigid", see e.g. perso.uclouvain.be/pierre-emmanuel.caprace/papers_pdf/… $\endgroup$– MishaCommented Dec 3, 2012 at 21:00
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$\begingroup$ @Jim actually the question is as follows,how the length function of elements of weyl group are related with respect to two different simple systems? $\endgroup$ Commented Dec 4, 2012 at 4:32
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The answer to the question in the title is "no". Let $W=S_3$. Then the transpositions $\{ (12), (23) \}$ form a simple system and $\{ (12), (13) \}$ form another. The length of $(13)$ is $3$ in the first system and $1$ in the second.
As to the question "is there any relation?": Well, they are equal mod $2$. Also, as Lee Mosher says, they are within a constant factor of each other; that fact is only interesting for infinite Coxeter groups.
I suspect that there isn't any more to say, but it is always hard to be sure that there isn't some other relation between two things.
answered Dec 3, 2012 at 14:55