Difference equations for graded characters from quantum cluster algebra

TL;DR

This paper introduces new $q$-difference operators acting on symmetric polynomials related to graded tensor products of current algebra modules, connecting quantum cluster algebras, integrable systems, and representation theory.

Contribution

It generalizes Macdonald raising operators and constructs a representation of the quantum $Q$-system as difference operators on graded characters.

Findings

Operators form a representation of the quantum $Q$-system.

Conserved quantities act as difference operators on characters.

Generalizes quantum Toda equations for tensor products of KR-modules.

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 ${\mathfrak g}[u]$ KR-modules \cite{FL} for ${\mathfrak g}=A_r$. These operators are generalizations of the Kirillov-Noumi \cite{kinoum} Macdonald raising operators, in the dual $q$-Whittaker limit $t\to\infty$. They form a representation of the quantum $Q$-system of type $A$ \cite{qKR}. 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 $U_q({\mathfrak sl}_{r+1})$, 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 \cite{Etingof}. We obtain a generalization of the latter for arbitrary tensor products of KR-modules.