By Bangming Deng

ISBN-10: 1607092050

ISBN-13: 9781607092056

The speculation of Schur-Weyl duality has had a profound impact over many components of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and offers an algebraic, instead of geometric, method of affine quantum Schur-Weyl thought. to start, quite a few algebraic constructions are mentioned, together with double Ringel-Hall algebras of cyclic quivers and their quantum loop algebra interpretation. the remainder of the publication investigates the affine quantum Schur-Weyl duality on 3 degrees. This comprises the affine quantum Schur-Weyl reciprocity, the bridging function of affine q-Schur algebras among representations of the quantum loop algebras and people of the corresponding affine Hecke algebras, presentation of affine quantum Schur algebras and the realisation conjecture for the double Ringel-Hall algebra with an evidence of the classical case. this article is perfect for researchers in algebra and graduate scholars who are looking to grasp Ringel-Hall algebras and Schur-Weyl duality.

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**Example text**

Thus, V W2 ⊆ W2 . Since 2 2 1 ∈ W2 , V ⊆ W2 and, hence, V = W2 . For m 0, applying Lusztig’s formulas km+2 s ks ; 0 t = v t (v − v −1 )km+1 s k−m−1 s ks ; 0 t = −v −t (v − v −1 )k−m s ks ;0 in [52, p. 278] yields k±m s t 2 therefore, W2 = W1 = V . ks ; 0 t +1 + v 2t km s ks ; 0 t +1 ∈ W1 , for any m ks ; 0 t + v −2t k−m+1 s and ks ; 0 t 0. Hence, W2 ⊆ W1 and, 48 2. 3. The integral 0-part U 0 is the Z-subalgebra of U(C∞ ) genks ;0 erated by K i±1 , K it ;0 , k±1 1. Hence, U 0 is a free s , 1 , for i ∈ I and s Z-module with basis {x y | x ∈ K, y ∈ M}.

1. 1) where ym± are Z-linear combinations of certain u ± A such that d(A) = mδ and M(A) are decomposable. Proof. 3), for each A ∈ + (n), a A ∈ Z is divisible by (v−v −1 )σ (A) , where σ (A) = 1 i n, j ∈Z ai, j . Note that σ ( A) equals the number of indecomposable summands in M(A). 2. , σ (A) > 1). Since subrepresentations and quotient representations of each indecomposable representation of (n) are again indecomposable, it follows that, for each A ∈ + (n), ± ± ± ± ± u± A I ⊆ I and I u A ⊆ I . It suffices to show that xm± ≡v −nm m (v − v −m n u± E ) l,l+mn l=1 mod (v − v −1 )2 I± .

D(A), 0), then D (n) admits a Z-grading D (n) = ⊕m∈Z D (n)m , and D (n) inherits a Z-grading D (n) = ⊕m∈Z D (n)m , where D (n)m = D (n) ∩ D (n)m . 4, it is clear that there is a graded algebra homomorphism D ,C (n) → D (n)C . This map is injective since it is so on every triangular component. Now, a dimensional comparison on homogeneous components shows that it is an isomorphism. 6. Let H (n)+ be the Z-submodule of D (n) spanned by u + A for A ∈ + (n). Then H (n)+ is a Z-algebra which is isomorphic to H (n).

### A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory by Bangming Deng

by Michael

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