Definitions of length function on a Weyl group Let $\Phi$ be an irreducible root system and $W$ the Weyl group of $\Phi$. Denote by $\Delta=\{\alpha_1, \alpha_2,\ldots,\alpha_l\}$ the corresponding base. 
Can anyone give me the standard definition of length function of an element of $W$? Furthermore, are there any upper-bound or lower-bound of lengths? For instance, let $w=r_{\beta_1}\ldots r_{\beta_k}$, $\beta_i \in \Phi$ be any element of $W$ then Carter's Lemma 2 gives the upper bound of length that $l(w) \le l$. However, if I follow Jim's definition of length function that involves the reduced expression of $w$ when expressed as composition of simple reflections, I haven't gotten any bounds of lengths. I'm aware of that simple reflections generate $W$ so that there is no conflict between definitions of length function discussing here. But I couldn't figure out the relationships between two definitions.
Any help would be much appreciated.
 A: Carter is studying the maximum number of reflections (w.r.t. hyperplanes having a root as its normal) needed to represent an element of the Weyl group as their product.
Humphreys' definition uses only simple reflections (IMHO this is a more common definition of the length of an element of a Weyl group). In that case the number of factors we can use is just the number of simple roots, i.e. equal to the rank of the root system. In Carter's problem he can use all the roots - a much larger number, so no wonder he can get away with shorter products.
The (Humphreys') length function of $W$ is bounded from above by the number of positive roots. That bound is attained by the longest element of $W$. For roots systems of classical types $A,B,C,D$ the number of positive roots is a quadratic polynomial of the rank of the root system. For comparison, in the case of $E_8$, the rank is $8$, but the number of positive roots is $120$. So when writing an element of $W(E_8)$ using simple reflections you may need up to $120$ factors. The average number you need is $60$ - exactly one half of the maximum. 
