Macdonald operators at infinity

Maxim L Nazarov; Evgeny Sklyanin

doi:10.1007/s10801-013-0477-2

Macdonald operators at infinity

Maxim L Nazarov, Evgeny Sklyanin

Mathematics

Research output: Contribution to journal › Article › peer-review

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 TeX 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.

Original language	English
Pages (from-to)	23-44
Number of pages	22
Journal	Journal of Algebraic Combinatorics
Volume	40
Issue number	1
Early online date	27 Sept 2013
DOIs	https://doi.org/10.1007/s10801-013-0477-2
Publication status	Published - Aug 2014

Keywords

Macdonald symmetric functions

Access to Document

10.1007/s10801-013-0477-2

http://arxiv.org/abs/1212.2960

Cite this

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title = "Macdonald operators at infinity",

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 TeX 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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AB - 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 TeX 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.

KW - Macdonald symmetric functions

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