Multiplicaton formulas and canonical basis for quantum affine gl_n

2016/08/04 by Du, Jie, Zhao, Zhonghua
#FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1608.01423

Abstract

We will give a representation-theoretic proof for the multiplication formula in the Ringel-Hall algebra \frak H_Δ(n) of a cyclic quiver Δ(n) given in \cite[Thm~4.5]DuFu2015quantum. As a first application, we see immediately the existence of Hall polynomials for cyclic quivers, a fact established in \citeGuo1995hallpoly and \citeRingel1993composition, and derive a recursive formula to compute them. We will further use the formula and the construction of certain monomial base for \mathfrak H_Δ(n) given in \citeDengDuXiao2007generic, together with the double Ringel--Hall algebra realisation of the quantum loop algebra U_v(gl_n) in \citeDengDuFu2012double, to develop some algorithms and to compute the canonical basis for U_v(gl_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 U_v(gl_n).

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