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

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.
Complete list of metadatas

Cited literature [18 references]  Display  Hide  Download

https://hal-cea.archives-ouvertes.fr/cea-01531314
Contributor : Emmanuelle de Laborderie <>
Submitted on : Thursday, June 1, 2017 - 3:13:25 PM
Last modification on : Wednesday, January 23, 2019 - 2:39:04 PM
Long-term archiving on: Wednesday, September 6, 2017 - 6:50:04 PM

File

1704.00154.pdf
Files produced by the author(s)

Identifiers

  • 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. 2017. ⟨cea-01531314⟩

Share

Metrics

Record views

123

Files downloads

112