Root systems of Coxeter groupsLet (W, R) be a Coxeter system, as defined in [6], and for all r, s E R let mr~ be the order of rs in W. Let H = { c~r I r E R ) denote the set of simple roots and V the R-vector space with basis H. Let { , ): V x V ~ R be a symmetric bilinear form which satisfies (c~,., c~) = -cos(Tr/m~s) for all r, s E R such that mr~ is finite, and (c~r, ~x~) _< -1 if m,.s is infinite; then r 9 v = v -2(v, c~)~x~ (for all r E R and v E V) determines a faithful action of W on V which preserves ( , ). We refer to such representations as standard geometric realizations of W.The set.~ = {w.c~r ] w E W, r C R} is called the root system of W in V. The subsets 4 )+ = {~,.~nA~a~. E ~b I A~ _> 0 for all r E R} and ~b-= {c~ E V t -a E 4 +} are the sets of positive roots and negative roots respectively. Define the support of ct = ~R A~ar E ~/i to be the set suppR(ct) = { r E R [ A,. r 0 }. For w E W define the length of w to be l(w) = rain{1 E N I w = vj ...rt for some rl,..
.,rz E R} (where N is the set of nonnegative integers), and let N(w) = { a E ~+ [ w. a E ~-}.We start with a proposition which lists an assortment of well known facts (see [6], for example). Proposition 1.1 (i) ~b = ~+ U ~-. (ii) IN(w)[ = l(w) for all w E W.(iii) Let rl,..., rn, s E R with l(rl ... r~s) < l(rl ... r~). There exists i C {1,..., n} such that T 1 9 9 9 rn8 : 7" 1 9 9 9 Ti-lri+ 1 . 9 . rn.iv) For all w E W and r E R, t(wr) = ~ l(w) + 1 if w . c~ E ~+, ll(w)-1 /fw 9 c~ E ~-. (v) For all c~ E ~b there is a uniquely defined element ra E W such that rc, = wrw-1 for all w E W, r E R with ~ = w . ~r. Moreover, rc~ . v = v -2(~ , v)ot for all v E V.
