Theta basis for reciprocal generalized cluster algebras

Theta basis for reciprocal generalized cluster algebras 

Cluster algebras are characterized by binomial exchange relations. A natural generalization of these algebras, introduced by Chekhov and Shapiro, relaxes this restriction and allows the exchange polynomials to have arbitrarily many terms. Following the work of Gross, Hacking, Keel, and Kontsevich, we give the construction of scattering diagrams for the subclass of generalized cluster algebras with reciprocal exchange coefficients. We then define the theta basis for these algebras and show that the fixed data of the left companion algebra is, up to isomorphism, Langlands dual to that of the right companion algebra (and vice versa). This is joint work with Man-Wai Cheug and Gregg Musiker.