2015/05/07 by Di Francesco, Philippe, Kedem, Rinat
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1505.01657
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 \citeFL for \mathfrak g=Ar. These operators are generalizations of the Kirillov-Noumi \citekinoum Macdonald raising operators, in the dual q-Whittaker limit t→∞. They form a representation of the quantum Q-system of type A \citeqKR. 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(\mathfrak 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 \citeEtingof. We obtain a generalization of the latter for arbitrary tensor products of KR-modules.