By Bangming Deng
The idea of Schur-Weyl duality has had a profound impact over many parts of algebra and combinatorics. this article is unique in respects: it discusses affine q-Schur algebras and provides an algebraic, instead of geometric, method of affine quantum Schur-Weyl conception. to start, a variety of algebraic buildings 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 position 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 explanation 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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Additional resources for A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory
Preliminaries (or [13, Th. 1) below. If a is not sincere, say ai = 0, then ua = j∈I, j =i v a j (1−a j ) ai−1 ai+1 u i−1 · · · u a11 u ann · · · u i+1 ∈ C (n). [a j ]! 1) Thus, H (n) is generated by u i and u a , for i ∈ I and sincere a ∈ NI . Indeed, this result can be strengthened as follows; see also [67, p. 421]. 3. The Ringel–Hall algebra H (n) is generated by u i and u mδ , for i ∈ I and m 1. Proof. Let H be the Q(v)-subalgebra generated by u i and u mδ for i ∈ I and m 1. To show H = H (n), it suffices to prove u a = u [Sa ] ∈ H for all a ∈ NI .
1) Thus, H (n) is generated by u i and u a , for i ∈ I and sincere a ∈ NI . Indeed, this result can be strengthened as follows; see also [67, p. 421]. 3. The Ringel–Hall algebra H (n) is generated by u i and u mδ , for i ∈ I and m 1. Proof. Let H be the Q(v)-subalgebra generated by u i and u mδ for i ∈ I and m 1. To show H = H (n), it suffices to prove u a = u [Sa ] ∈ H for all a ∈ NI . Take an arbitrary a ∈ NI . We proceed by induction on σ (a) = i∈I ai to show u a ∈ H . If σ (a) = 0 or 1, then clearly u a ∈ H .
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± . We proceed by induction on m. If m = 1, it is trivial since x1± = c1± . Let now m > 1. 2) implies that xm± ≡ mv (1−n)m−1 (v − v −1 ) n m−1 u± E l,l+mn l=1 n × v (1−n)(m−s)−1 (v − v −1 ) l, l u± E l,l+sn v −ns (v s − v −s ) s=1 u± E l,l+(m−s)n l=1 It is clear that for 1 − n u± E l,l+sn l=1 mod (v − v −1 )2 I± . n, u± E ≡ δl,l u ± E l ,l +(m−s)n l,l+mn mod I± . We conclude that xm± ≡ v (1−n)m m(1 − v −2 ) − m−1 s=1 ≡ v −nm (v m − v −m ) n u± E l=1 n (1 − v −2 )(1 − v −2s ) l=1 l,l+mn u± E l,l+mn mod (v − v −1 )2 I± .
A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory by Bangming Deng