Categorification of tensor powers of the vector by Antonio Sartori

By Antonio Sartori

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Then the translation functors Tλµ corresponds to • ⊗Aµ Tλµ P(µ) ∼ = • ⊗ Aµ λ HomO (P(λ), Tµ P(µ)). Let now λ, µ ∈ ✄+ with Sµ ⊆ Sλ . 5) Tλµ corresponds to • ⊗Aµ HomB λ VP(λ), B λ ⊗B µ VP(µ) . Hence these functors are gradable. 7) Tλµ = • ⊗Aµ HomB λ VP(λ), B λ ⊗B µ VP(µ) − (x0 ) , where x0 is the longest element in Sµ Sλ and Tλµ M (µ) = P (x0 · λ). short . 1. 8) Tλµ q (x0 ) Tµλ and Tµλ q− (x0 ) Tλµ . Chapter 4. Graded category O 45 Proof. 8) on the dominant Verma modules. 10) C = Hom(P (µ), M (µ)) = Hom(Tµλ M (λ), M (µ)) = Hom(M (λ), q − (x0 ) Tλµ M (µ)) = Hom(M (λ), q − (x0 ) P (x0 · λ)) = C.

3) ˇ 2 = (q −1 − q)H ˇ + Id. 4) ˇ i = id⊗i−1 ⊗ H ˇ ⊗ id⊗n−i−1 . H By definition, they are intertwiners for the action of Uq . 1. 3), and the endomorˇ is just the inverse of the braiding R ˇ V,V . From this it follows directly that H ˇ is equivphism H ˇ ˇ ˇ ˇ ˇ ˇ ariant and that the braid relation Hi Hi+1 Hi = Hi+1 Hi Hi+1 holds for all i = 1, . . , n − 1. The following result is also known as super Schur-Weyl duality. The non-quantized version was originally proved by Berele and Regev ([BR87]) and independently by Sergeev ([Ser84]).

N ··· 1 1 ··· 1 Chapter 3. Graphical calculus for gl(1|1)–representations 37 Proof. 3) that the element H wn acts by 0 on Mpq unless Wp is trivial. 13) it follows that H wn acts by 0 on the weight space (V ⊗n )k unless k = n − 1, n. 16) and the one copy of L(nε1 ) is the unique summand whose weight spaces are contained in (V ⊗n )n−1 and (V ⊗n )n . It follows that, up to a multiple, H wn acts on V ⊗n by projecting onto this summand L(nε1 ). 24) coincide up to a multiple. H wn . 2b) the same holds for Ξn , hence the claim follows.

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