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. [6] 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. [7] Roger Carter and M. T. J. Payne, On homomorphisms between Weyl modules and Specht modules, Math. Proc. Camb. Phil. Soc. 87 (1980), 419–425. [8] 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.