($${{\mathbf {t}}},{{\mathbf {q}}}$$)-Deformed Q-Systems, DAHA and Quantum Toroidal Algebras via Generalized Macdonald Operators

Abstract

We introduce the natural (t, q)-deformation of the Q-system algebra in type A. The q-Whittaker limit $$t\rightarrow \infty $$ gives the quantum Q-system algebra of Di Francesco and Kedem (Lett Math Phys 107(2):301–341, [DFK17]), a deformation of the Groethendieck ring of finite dimensional Yangian modules, compatible with graded tensor products (Hatayama et al. in: Recent Developments in Quantum Affine Algebras and Related Topics (Raleigh, NC, 1998), Volume 248 of Contemporary Mathematics, Amer. Math. Soc., Providence, [HKO+99]; Feigin and Loktev in: Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Volume 194 of Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, [FL99]; Di Francesco and Kedem in Int Math Res Not IMRN 10:2593–2642, [DFK14]). We show that the (q, t)-deformed algebra is isomorphic to the spherical double affine Hecke algebra of type $${\mathfrak {gl}}_N$$ . Moreover, we describe the kernel of the surjective homomorphism from the quantum toroidal algebra (Miki in J Math Phys 48(12):123520, [Mik07]) and the elliptic Hall algebra (Schiffmann and Vasserot in Compos Math 147(1):188–234, [SV11]) to this new algebra. It is generated by (q, t)-determinants, new objects which are a deformation of the quantum determinant associated with the quantum Q-system. The functional representation of the algebra is generated by generalized Macdonald operators, obtained from the usual Macdonald operators by the $$SL_2({\mathbb {Z}})$$ -action on the spherical Double Affine Hecke Algebra. The generating function for generalized Macdonald operators acts by plethysms on the space of symmetric functions. We give the relation to the plethystic operators from Macdonald theory of Bergeron et al. (J Comb 7(4):671–714, [BGLX16]) in the limit $$N\rightarrow \infty $$ . Thus, the (q, t)-deformation of the Q-system cluster algebra leads directly to Macdonald theory.