# (t,q) Q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators

* Auteur correspondant
Abstract : We introduce difference operators on the space of symmetric functions which are a natural generalization of the $(q,t)$-Macdonald operators. In the $t\to\infty$ limit, they satisfy the $A_{N-1}$ quantum $Q$-system. We identify the elements in the spherical $A_{N-1}$ DAHA which are represented by these operators, as well as within the quantum toroidal algebra of $gl_1$ and the elliptic Hall algebra. We present a plethystic, or bosonic, formulation of the generating functions for the generalized Macdonald operators, which we relate to recent work of Bergeron et al. Finally we derive constant term identities for the current that allow to interpret them in terms of shuffle products. In particular we obtain in the $t\to\infty$ limit a shuffle presentation of the quantum $Q$-system relations.
Type de document :
Pré-publication, Document de travail
t17/082. 57 pages. 2017
Domaine :

Littérature citée [18 références]

https://hal-cea.archives-ouvertes.fr/cea-01531314
Contributeur : Emmanuelle De Laborderie <>
Soumis le : jeudi 1 juin 2017 - 15:13:25
Dernière modification le : mercredi 23 janvier 2019 - 14:39:04
Document(s) archivé(s) le : mercredi 6 septembre 2017 - 18:50:04

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1704.00154.pdf
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• HAL Id : cea-01531314, version 1
• ARXIV : 1704.00154

### Citation

Philippe Di Francesco, Rinat Kedem. (t,q) Q-systems, DAHA and quantum toroidal algebras via generalized Macdonald operators. t17/082. 57 pages. 2017. 〈cea-01531314〉

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