Combinatorics of Coxeter Groups (Graduate Texts in by Anders Bjorner, Francesco Brenti

By Anders Bjorner, Francesco Brenti

Contains a wealthy number of workouts to accompany the exposition of Coxeter teams Coxeter teams have already been exposited from algebraic and geometric views, yet this publication should be featuring the combinatorial elements of Coxeter teams

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Sj1 . . sj2 . . sjk . . sq . Assume that ik < jk , and let tjk = sq sq−1 . . sjk . . sq−1 sq . Then, xk−1 = utjk = s1 . . si1 . . si2 . . sik . . sjk . . sq , so (xk−1 ) ≤ (u) − 1, contradicting that xk−1 ✄ u. Hence, jk ≤ ik , and by symmetry, ik ≤ jk . The equality ik = jk implies that xk−1 = xk−1 . Since the interval [xk−1 , w] by the induction assumption does not admit two distinct maximal chains with increasing labels, we conclude that m = m . ✷ From this, we can deduce the structure of intervals of length 2.

Then, w = s s1 s2 . . sq is also reduced, and there exists a reduced subword u = si1 si2 . . sik ≺ s s1 s2 . . sq . Now, si1 = s since su > u; hence, si1 si2 . . sik ≺ s1 s2 . . sq ⇒ u ≤ sw and s si1 si2 . . sik ≺ ss1 s2 . . sq ⇒ su ≤ w. 8 stw. (i) For s ∈ S, t ∈ T , s = t: w ✁ sw, tw ⇒ sw, tw ✁ 36 2. Bruhat order (ii) For s, s ∈ S: w✁sw, ws ⇒ either sw, ws ✁sws or w = sws . ✷ Recall that a poset P is said to be directed if for any u, w ∈ P , there exists z ∈ P such that u, w ≤ z. 9 Bruhat order is a directed poset.

Sik . . sid . . sr . . sid . . sik . . sq , for some r < ik , r = ij . 9) In the first case, w = w t2 = (s1 s2 . . sq )(sq . . sik . . sq )(sq . . sp . . sq ) = s1 . . sik . . sp . . sq , 34 2. Bruhat order which contradicts (w) = q. Similarly, in the second case, u = u t2 = (s1 . . si1 . . sik . . sq )(sq . . sik . . sr . . sik . . sq )(sq . . sik . . sq ) = s1 . . si1 . . sr . . sik . . sq , which contradicts the minimality of ik . 2 (Subword Property) Let w = s1 s2 .

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