### Abstract

We introduce a new set of q-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra g[ u] KR-modules [FL1] for g= A
_{r}
. These operators are generalizations of the Kirillov-Noumi [KN] Macdonald raising operators, in the dual q-Whittaker limit t → ∞. They form a representation of the quantum Q-system of type A, see [DFK3]. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of U
_{q}
(sl
_{r}
_{+}
_{1}
) , act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I q-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations [E]. For other modules, the action of the conserved quantities is thus a generalization of these difference equations in the case of an arbitrary (higher level) tensor products of KR-modules.

Original language | English (US) |
---|---|

Pages (from-to) | 391-424 |

Number of pages | 34 |

Journal | Transformation Groups |

Volume | 23 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 2018 |

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### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

**DIFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA.** / Di Francesco, P.; Kedem, R.

Research output: Contribution to journal › Article

*Transformation Groups*, vol. 23, no. 2, pp. 391-424. https://doi.org/10.1007/s00031-018-9480-y

}

TY - JOUR

T1 - DIFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA

AU - Di Francesco, P.

AU - Kedem, R.

PY - 2018/6/1

Y1 - 2018/6/1

N2 - We introduce a new set of q-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra g[ u] KR-modules [FL1] for g= A r . These operators are generalizations of the Kirillov-Noumi [KN] Macdonald raising operators, in the dual q-Whittaker limit t → ∞. They form a representation of the quantum Q-system of type A, see [DFK3]. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of U q (sl r + 1 ) , act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I q-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations [E]. For other modules, the action of the conserved quantities is thus a generalization of these difference equations in the case of an arbitrary (higher level) tensor products of KR-modules.

AB - We introduce a new set of q-difference operators acting as raising operators on a family of symmetric polynomials which are characters of graded tensor products of current algebra g[ u] KR-modules [FL1] for g= A r . These operators are generalizations of the Kirillov-Noumi [KN] Macdonald raising operators, in the dual q-Whittaker limit t → ∞. They form a representation of the quantum Q-system of type A, see [DFK3]. This system is a subalgebra of a quantum cluster algebra, and is also a discrete integrable system whose conserved quantities, analogous to the Casimirs of U q (sl r + 1 ) , act as difference operators on the above family of symmetric polynomials. The characters in the special case of products of fundamental modules are class I q-Whittaker functions, or characters of level-1 Demazure modules or Weyl modules. The action of the conserved quantities on these characters gives the difference quantum Toda equations [E]. For other modules, the action of the conserved quantities is thus a generalization of these difference equations in the case of an arbitrary (higher level) tensor products of KR-modules.

UR - http://www.scopus.com/inward/record.url?scp=85044932637&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85044932637&partnerID=8YFLogxK

U2 - 10.1007/s00031-018-9480-y

DO - 10.1007/s00031-018-9480-y

M3 - Article

AN - SCOPUS:85044932637

VL - 23

SP - 391

EP - 424

JO - Transformation Groups

JF - Transformation Groups

SN - 1083-4362

IS - 2

ER -