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# Corollary 844 if and is not then the root string

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Unformatted text preview: ght zero part of W is W0 = F hα . So, W contains only one irreducible V (even). Since we know that W contains Sα ∼ V (2), there cannot be any = other V (even). So, L2α = 0. In other words, twice a root cannot be a root. But then 1 α 2 also cannot be a root since twice of it is a root. And this implies that W1 = Lα/2 = 0. So, W = V (2) is irreducible. This also implies that Lα is one dimensional. ￿ Corollary 8.4.4. If α, β ∈ Φ and β is not ±α then the α-root string through β is irreducible and has the form: V (m) ∼ Lβ +qα ⊕ Lβ +(q−1)α ⊕ · · · ⊕ Lβ −rα = and m = q + r. And β (hα ) = q − r is an integer. (β (hα ) are the Cartan integers.) Proof. Let M be the α-root string through β . Then M0 = 0 since the root string does not go through 0. (The proof of the last proposition showed that Sα = Lα ⊕ F hα ⊕ L−α is the only α-root string through 0.) Therefore, M is a direct sum of V (odd)s. So, the α-root string has only Modd but a diﬀerence of 2 in hα -weights corresponds to a diﬀerence of α in roots since α(hα ) = 2. So, the root string contains only Lβ +kα for integer k . One of these is M1 and thus is 1-dimensional. So, M is irreducible. If M ∼ V (m) then Mm = Lβ +qα and M−m = Lβ −rα . The dimension of M is q + r + 1 = = m + 1. This implies that q + r = m = (β + q α)(hα ) = β (hα ) + 2q ￿ So, β (hα ) = r − q ∈ Z. 8.5. Inner product on H ∗ . We just proved that β (hα ) ∈ Z. We will rephrase this in terms of the inner product on H ∗ . t Recall that hα = κ(t2αα α ) . And,...
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## This document was uploaded on 03/09/2014 for the course MATH 223a at Brandeis.

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