By Nerode A., Shore R. (eds.)
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In particular all roots have the same length and all dβ are 1. 2. Let β be an arbitrary positive root. Then for any λ ∈ X + for which λ − β is also in X + we have ExtiCA (∇A (λ), ∇A (λ − β)) A/([ λ + ρ, β ∨ − 1])A 0 if i = 1, otherwise. 2. Adjacent Weyl modules. Let R again be arbitrary. Denote by A(v−q) the localization of A at the maximal ideal generated by v − q. 1. 3. Suppose l ≥ h. Let (λ, λ ) be a pair of adjacent l-regular dominant weights separated by the hyperplane Hβ,n and with λ > λ .
104 (1991), 1–59.  Henning Haahr Andersen, Jens Carsten Jantzen and Wolfgang Soergel, Representations of quantum groups at a p−th root of unity and of semisimple groups in characteristic p: Independence of p, Asterisque 220 (1994), pp. 1–321.  Roger Carter and M. T. J. Payne, On homomorphisms between Weyl modules and Specht modules, Math. Proc. Camb. Phil. Soc. 87 (1980), 419–425.  Stephen Donkin, Rational representations of algebraic groups, Lecture Notes in Mathematics 1140 (Springer 1985).
We ﬁrst observe that yλ acts as zero on any simple tensor vi1 ⊗ · · · ⊗ vir ∈ V ⊗r , for any λ ∈ P . ,λ } (sgn w)w. , we have yλ = αβ for some β ∈ KSr , so yλ acts as zero as well. Since V ⊗r is spanned by such simple tensors, it follows that yλ acts as zero on V ⊗r . It follows immediately that every yST = d(S)−1 yλ d(T ), for λ ∈ P , acts as zero on V ⊗r , for any λ-tableaux S, T , since d(S)−1 simply permutes the entries in the tensor, and then yλ annihilates it. Since A[P ] is spanned by such yST , it follows that A[P ] is contained in the kernel of Φ.