An alternative polynomial representation of spherical DAHA

Seminar: 
Clusters and Geometry
Event time: 
Friday, October 30, 2020 - 1:00pm
Speaker: 
Gus Schrader
Speaker affiliation: 
Columbia University
Event description: 

The spherical subalgebra of Cherednik's double affine Hecke algebra of type A has a polynomial representation in which the algebra acts on a space of symmetric Laurent polynomials by rational q-difference operators. This representation has many useful applications e.g. to the theory of Macdonald polynomials. I'll present an alternative polynomial representation of the spherical DAHA, in which the algebra acts on a space of non-symmetric Laurent polynomials by Laurent polynomial q-difference operators. This latter representation turns out to be compatible with a natural cluster algebra structure, in such a way that the action of the modular group on DAHA is given by cluster transformations. Based on joint work in progress with Philippe di Francesco, Rinat Kedem, and Alexander Shapiro.