Macdonald operators at infinity |
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Authors: | M L Nazarov E K Sklyanin |
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Institution: | 1. Department of Mathematics, University of York, York, YO10 5DD, UK
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Abstract: | We construct a family of pairwise commuting operators such that the Macdonald symmetric functions of infinitely many variables x 1,x 2,… and of two parameters q,t are their eigenfunctions. These operators are defined as limits at N→∞ of renormalized Macdonald operators acting on symmetric polynomials in the variables x 1,…,x N . They are differential operators in terms of the power sum variables \(p_{n}=x_{1}^{n}+x_{2}^{n}+\cdots\) and we compute their symbols by using the Macdonald reproducing kernel. We express these symbols in terms of the Hall–Littlewood symmetric functions of the variables x 1,x 2,…. Our result also yields elementary step operators for the Macdonald symmetric functions. |
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