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Multiplication formulas and canonical bases for quantum affine gln

  • Jie Du,
    School of Mathematics and Statistics, University of New South Wales, Sydney 2052, Australia
  • Zhonghua Zhao,
    School of Science, Beijing University of Chemical Technology, Beijing 100029, China
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Abstract

We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra $\mathfrak{H}_\Delta(n)$ of a cyclic quiver $\Delta(n)$. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established by J. Y. Guo and C. M. Ringel, and derive a recursive formula to compute them. We will further use the formula and the construction of a certain monomial base for $\mathfrak{H}_\Delta(n)$ given by Deng, Du, and Xiao together with the double Ringel--Hall algebra realisation of the quantum loop algebra $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_n)$ given by Deng, Du, and Fu to develop some algorithms and to compute the canonical basis for $\mathbf{U}_v^+(\widehat{\mathfrak{g}\mathfrak{l}}_n)$. As examples, we will show explicitly the part of the canonical basis associated with modules of Lowey length at most $2$ for the quantum group $\mathbf{U}_v(\widehat{\mathfrak{g}\mathfrak{l}}_2)$.
Keywords: Ringel-Hall algebra, quantum group, cyclic quiver, monomial basis, canonical basis Ringel-Hall algebra, quantum group, cyclic quiver, monomial basis, canonical basis
MSC Classifications: 16G20, 20G42 show english descriptions Representations of quivers and partially ordered sets
Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]
16G20 - Representations of quivers and partially ordered sets
20G42 - Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]
 

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