Abstract
Let [Formula: see text] be the modified quantum affine [Formula: see text] and let [Formula: see text] be the positive part of quantum affine [Formula: see text]. Let [Formula: see text] be the canonical basis of [Formula: see text] and let [Formula: see text] be the canonical basis of [Formula: see text]. In this paper, we use the theory of affine quantum Schur algebras to prove that the structure constants for the comultiplication with respect to [Formula: see text] are determined by the structure constants for the comultiplication with respect to [Formula: see text] for [Formula: see text]. In particular, from the positivity property for the comultiplication of [Formula: see text], we obtain the positivity property for the comultiplication of [Formula: see text], which is conjectured by Lusztig [Introduction to Quantum Groups, Progress in Mathematics, Vol. 110 (Birkhäuser, Boston, 1993), 25.4.2].
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