TY - JOUR

T1 - DIFFERENCE EQUATIONS FOR GRADED CHARACTERS FROM QUANTUM CLUSTER ALGEBRA

AU - Di Francesco, P.

AU - Kedem, R.

N1 - Funding Information:
Acknowledgments. We thank O. Babelon, M. Bergvelt, A. Borodin, I. Chered-nik, I. Corwin, V. Pasquier, and S. Shakirov for discussions at various stages of this work. R.K.’s research is supported by NSF grant DMS-1404988. P.D.F. is supported by the NSF grant DMS-1301636 and the Morris and Gertrude Fine endowment. R.K. would like to thank the Institut de Physique Théorique (IPhT) of Saclay, France, for hospitality during various stages of this work.

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 Uq(slr + 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 Uq(slr + 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.

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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 -