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?



length function of a coxeter group with respect to two different simple systems are equal or not?
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(?) 


