Difference equations for graded characters from quantum cluster algebra

Philippe Di Francesco (1) , Rinat Kedem (2)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681
2 Department of Mathematics [Urbana]

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 = Ar. 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

Domains: Mathematical Physics [math-ph]

Philippe Di Francesco, Rinat Kedem. Difference equations for graded characters from quantum cluster algebra. Transformation Groups, 2018, 23, pp.391-424. ⟨10.1007/s00031-018-9480-y⟩. ⟨cea-01251612⟩