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A Double Hall Algebra Approach to Affine Quantum Schur-Weyl Theory by Qiang Fu, Jie Du, Bangming Deng

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3

Affine quantum Schur algebras and the Schur–Weyl reciprocity

Like the quantum Schur algebra, the affine quantum Schur algebra has several equivalent definitions. We first present the geometric definition, given by Ginzburg–Vasserot and Lusztig, which uses cyclic flags and the convolution product. We then discuss the two Hecke algebra definitions given by R. Green and by Varagnolo–Vasserot. The former uses q-permutation modules, while the latter uses tensor spaces. Both versions are related by the Bernstein presentation for Hecke algebras of affine type A.

In §3.4, we review the construction of BLM type bases for affine quantum Schur algebras and the multiplication formulas between simple generators and BLM basis elements (Theorem 3.4.2) developed ...

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