TY - JOUR

T1 - (t, q)-Deformed Q-Systems, DAHA and Quantum Toroidal Algebras via Generalized Macdonald Operators

AU - Di Francesco, Philippe

AU - Kedem, Rinat

N1 - Funding Information:
Acknowledgments. We thank O. Babelon, F, Bergeron, J.-E. Bourgine, I. Cherednik, M. Jimbo, Y. Matsuo, A. Negut, V. Pasquier, O. Schiffmann for valuable discussions. We thank the Institut de Physique Théorique (IPhT) of CEA-Saclay, France, and the Institut Henri Poincaré, Paris 2017 program on “Combinatorics and Interactions”, France for hospitality during various stages of this work. This research was supported by NSF Grants DMS-1301636, DMS-1404988, and DMS–1802044. P.D.F. is supported by the Morris and Gertrude Fine Foundation.
Funding Information:
We thank O. Babelon, F, Bergeron, J.-E. Bourgine, I. Cherednik, M. Jimbo, Y. Matsuo, A. Negut, V. Pasquier, O. Schiffmann for valuable discussions. We thank the Institut de Physique Th?orique (IPhT) of CEA-Saclay, France, and the Institut Henri Poincar?, Paris 2017 program on ?Combinatorics and Interactions?, France for hospitality during various stages of this work. This research was supported by NSF Grants DMS-1301636, DMS-1404988, and DMS?1802044. P.D.F. is supported by the Morris and Gertrude Fine Foundation.
Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - We introduce the natural (t, q)-deformation of the Q-system algebra in type A. The q-Whittaker limit t→ ∞ 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 glN. 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 SL2(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→ ∞. Thus, the (q, t)-deformation of the Q-system cluster algebra leads directly to Macdonald theory.

AB - We introduce the natural (t, q)-deformation of the Q-system algebra in type A. The q-Whittaker limit t→ ∞ 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 glN. 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 SL2(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→ ∞. Thus, the (q, t)-deformation of the Q-system cluster algebra leads directly to Macdonald theory.

UR - http://www.scopus.com/inward/record.url?scp=85067291071&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067291071&partnerID=8YFLogxK

U2 - 10.1007/s00220-019-03472-x

DO - 10.1007/s00220-019-03472-x

M3 - Article

AN - SCOPUS:85067291071

VL - 369

SP - 867

EP - 928

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -