By M. Raynaud, T. Shioda
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Additional resources for Algebraic Geometry. Proc. conf. Tokyo, Kyoto, 1982
Any element D of W is thus a word µ1 · · · µl(D) in the letters from the alphabet ∪ − , where l(D) is its length. For α ∈ we shall denote by α¯ the opposite element from − . 4 The braid and Hecke semigroups Let B be the quotient of the semigroup W by αβα = βαβ αβαβ = βαβα αβαβαβ = βαβαβα ¯ α β¯ = βα, and α¯ β¯ α¯ = β¯ α¯ β¯ and α¯ β¯ α¯ β¯ = β¯ α¯ β¯ α¯ ¯ and α¯ β α¯ β¯ α¯ β¯ = β¯ α¯ β¯ α¯ β¯ α¯ (1) if Cαβ = Cβα = −1, if Cαβ = 2Cβα = −2, (2) if Cαβ = 3Cβα = −3. The semigroup B is called the braid semigroup.
One has H α (x) = exp(log(x)hα ). Let us introduce the group elements Eα = exp eα , Fα = exp fα . Replace the letters in D by the group elements using the rule α → Eα , α¯ → α F . For α ∈ and for any (αi ) ∈ J (D), insert H α (xiα ) somewhere between the corresponding walls. The choice in placing every H is nonessential since it commutes with all Es and F s unless they are marked by the same root. In other words the sequence of group elements is deﬁned by the following requirements: • • • • The sequence of Es and Fs reproduce the sequence of letters in the word D.
Moreover, for every γ ∈ , the subset of γ -decorated elements of J (D) has a natural linear order, induced by the ones on J (αi ), and the linear order of the word D. 2. , minimal and maximal) elements for the deﬁned above linear order on the γ -decorated part of J (D). The seed J(D) is obtained from the seed J(D) by reducing J0 (D) to the subset J0 (D). Observe that εαβ is integral unless both α and β are in J0 (D). Thus the integrality condition for εαβ holds. 3 An alternative deﬁnition of the seed J(D) The sets J0 (D) and J (D) Given a positive simple root α ∈ α and α¯ in the word D.
Algebraic Geometry. Proc. conf. Tokyo, Kyoto, 1982 by M. Raynaud, T. Shioda