For Coxeter group theory and its applications, we refer the reader to the books [1], [5], [8], [10] (and references cited there).
Testo completo
By a subword of a word s 1 -s 2 - · · · -s q (where we use the symbol “-” to separate letters in a word in the alphabet S) we mean a word of the form s i1
with the dihedral parabolic subgroup W {r,r0
Let M 0 be the matrix whose entry m 0 (r, r 0 ) is `(w 0 (r, r 0 )), the length of the element w 0 (r, r 0 ), for each pair (r, r 0 ) ∈ S × S. Let (W 0 , S) be the Coxeter system with the same set S of Coxeter generators but having M 0 as Coxeter matrix. Then all reduced expressions of the element w ∈ W are also reduced expressions as expressions in W 0 and are reduced expressions of a unique element, say w 0 ∈ W 0 . By the Subword Property (Theorem 2.2), the intervals [e, w] ⊆ W and [e, w 0 ] ⊆ W 0 are isomorphic as posets and hence the special matchings of w and those of w 0 correspond. This is the reason why, in the study of the special matchings of w, it would be natural to assume that, for all r, r 0 ∈ S, the Coxeter matrix entry m r,r0
(2) Let R = (I, s, t, ρ s ) and R 0 = (I 0 , s, t 0 , ρ s ) be right systems for w. Then M R = M R0
We now prove (2). If I ∪ C s = I 0 ∪C s we have M R = M R0
Remark 5.3. A word of caution is needed. If I, I 0 ⊆ S are such that s ∈ I ∩ I 0 , t / ∈ I ∩ I 0 and I ∪ C s = I 0 ∪ C s then it is not true that R = (I, s, t, ρ s ) is a right system if and only if R 0 = (I 0 , s, t, ρ s ) is. This fails, e.g., in A 4 with w = s 4 s 2 s 3 s 2 s 1 , s = s 3 , t = s 4 , I = {s 1 , s 2 , s 3 } and I 0 = {s 2 , s 3 }. In this case we have I ∪ C s = I 0 ∪ C s = {s 1 , s 2 , s 3 } and so K(I) = K(I 0 ) = {s 1 , s 3 , s 4 }. Then we can observe that w I = s 4 ∈ W K and in fact one can verify that the map u 7→ u I s 2 u I defines a special matching of w. On the other hand w I0
{M J,s with s ∈ S, C s ⊆ J ⊆ S, w J ∈ W Cs
M R (u) = aββM st (v 2 )v 3 = aM st (v 2 )v 3
(3) if R = (J, s, t, M st ) and R 0 = (J 0 , s 0 , t 0 , M s0
• M st (u) = us for all u ≤ w 0 (s, t) and M s0
• t = t 0 and M st = M s0
(4) if R = (J, s, t, M st ) is a right system and L = (K, s 0 , t 0 , M s0
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