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Definition A Coxeter system (W, S) consists of a group W (called a Coxeter group) generated by a set S of involutions with presentation W = 〈S: s2 = 1, (st)m(s,t) = 1〉, where m(s, t) ≥ 2 for s 6 = t. Comment Since s and t are involutions, the relation (st)m(s,t) = 1 can be rewritten as m(s, t) = 2 = ⇒ st = ts} short braid relations m(s, t) = 3 = ⇒ sts = tst m(s, t) = 4 = ⇒ stst = tsts long braid relations D.C. Ernst On the CFC elements of Coxeter groups 2 / 1