Length of a Weyl group element

In mathematics, the length of an element "w" in a Weyl group "W", denoted by "l"("w"), is the smallest number "k" so that "w" is a product of "k" reflections by simple roots. (So, the notion depends on the choice of a positive Weyl chamber.) In particular, a simple reflection has length one. The function "l" is then an integer-valued function of "W"; it is a length function of "W". It follows immediately from the definition that "l"("w"−1) = "l"("w") and that "l"("ww"'−1) ≤ "l"("w") + "l"("w' ").