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We will construct the Lusztig form for the quantum loop algebra of $\mathfrak{gl}_n$ by proving the conjecture \cite[3.8.6]{DDF} and establish partially the Schur--Weyl duality at the integral level in this case.
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1999
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M. Varagnolo and E. Vasserot, On the decomposition matrices of the quantized Schur algebra
1999
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O. Schiffmann and E. Vasserot, Geometric construction of the global base of the quantum modified algebra of 𝔤 𝔩 ^ n \widehat{\mathfrak{gl}}_{n}
2000
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M. Reineke, Generic extensions and multiplicative bases of quantum groups at q = 0 q=0
2001
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2010
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B. Deng, J. Du and Q. Fu, A double Hall algebra approach to affine quantum Schur–Weyl theory
2012
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K. Mcgerty, On the geometric realization of the inner product and canonical basis for quantum affine 𝔰 𝔩 n \mathfrak{sl}_{n}
2012
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Q. Fu, Integral affine Schur–Weyl reciprocity
2013
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Q. Fu, Canonical bases for modified quantum 𝔤 𝔩 n \mathfrak{gl}_{n} and q q -Schur algebras
2014
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