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Thursday, September 14, 2017

**Abstract:** We introduce a generalization of the A-type Macdonald difference operators via a symmetrization identity (S) that maps symmetric functions to difference operators. Recall that Macdonald operators have Macdonald polynomials as common eigenfunctions, and play a crucial role in deep combinatorial theorems (factorial n theorem, shuffle theorem, general questions about Schur positivity). We will show how our generalized Macdonald operators can be obtained by implementing the modular group symmetry of the A-type Double Affine Hecke Algebra in its functional representation. The corresponding difference operators obey commutation relations that can be viewed as t-deformations of some particular quantum cluster algebra relations pertaining to the A-type Q-system. We’ll show how these can be canned into a representation of the affine gl1 quantum toroidal algebra with zero central charge. We finally present an alternative definition of our generalized Macdonald difference operators via a constant term identity (CT) that maps symmetric functions to difference operators. This leads to a natural definition of Shuffle product, for which (S) and (CT) are algebra morphisms. We then show how commutation relations for our operators reduce to simple Shuffle identities. This is illustrated in the quantum cluster algebra limit t->infinity.

Thursday, October 19, 2017