DIFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)391-424
Number of pages34
JournalTransformation Groups
Volume23
Issue number2
DOIs
StatePublished - Jun 1 2018

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Cluster Algebra
Quantum Algebra
Difference equation
Conserved Quantity
Module
Symmetric Polynomials
Difference Operator
Tensor Product
Operator
Weyl Modules
Whittaker Function
Current Algebra
Integrable Systems
Discrete Systems
Subalgebra
Character
Arbitrary

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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

In: Transformation Groups, Vol. 23, No. 2, 01.06.2018, p. 391-424.

Research output: Contribution to journalArticle

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