Sekiguchi-Debiard operators at infinity. (English) Zbl 1408.81016

Nazarov, M. L.; Sklyanin, E. K.

Commun. Math. Phys. 324, No. 3, 831-849 (2013).

Summary: We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables \(x_1, x_2, \dots\) are their eigenfunctions. These operators are defined as limits at \(N \rightarrow \infty \) of renormalised Sekiguchi-Debiard operators acting on symmetric polynomials in the variables \(x_1, \dots, x_N\). They are differential operators in terms of the power sum variables \(p_n = x^n_1 + x^n_2 + \cdots\) and we compute their symbols by using the Jack reproducing kernel. Our result yields a hierarchy of commuting Hamiltonians for the quantum Calogero-Sutherland model with an infinite number of bosonic particles in terms of the collective variables of the model. Our result also yields the elementary step operators for the Jack symmetric functions.

MSC:

81R15	Operator algebra methods applied to problems in quantum theory
33C20	Generalized hypergeometric series, \({}_pF_q\)
33C52	Orthogonal polynomials and functions associated with root systems
05E05	Symmetric functions and generalizations
37K10	Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
81V70	Many-body theory; quantum Hall effect

References:

[1]	Abanov A.G., Wiegmann P.B.: Quantum Hydrodynamics, the quantum Benjamin-Ono equation, and the Calogero model. Phys. Rev. Lett. 95, 076402 (2005) · doi:10.1103/PhysRevLett.95.076402
[2]	Awata H., Kanno H.: Macdonald operators and homological invariants of the colored Hopf link. J. Phys. A 44, 375201 (2011) · Zbl 1235.81105
[3]	Awata H., Matsuo Y., Odake S., Shiraishi J.: Collective fields, Calogero-Sutherland model and generalized matrix models. Phys. Lett. B 347, 49-55 (1995) · Zbl 0894.17027
[4]	Cai W., Jing N.: Applications of Laplace -Beltrami operator for Jack polynomials. Eur. J. Comb. 33, 556-571 (2012) · Zbl 1298.05318 · doi:10.1016/j.ejc.2011.11.003
[5]	Calogero F.: Ground state of a one-dimensional N-body system. J. Math. Phys. 10, 2197-2200 (1969) · doi:10.1063/1.1664821
[6]	Debiard A.: Polynômes de Tchébychev et de Jacobi dans un espace euclidien de dimension p. C. R. Acad. Sc. Paris I 296, 529-532 (1983) · Zbl 0557.33004
[7]	Khoroshkin S., Nazarov M., Papi P.: Irreducible representations of Yangians. J. Algebra 346, 189-226 (2011) · Zbl 1263.17015 · doi:10.1016/j.jalgebra.2011.08.011
[8]	Lapointe L., Vinet L.: Exact operator solution of the Calogero-Sutherland model. Commun. Math. Phys. 178, 425-152 (1996) · Zbl 0859.35103
[9]	Macdonald I.G.: Symmetric Functions and Hall Polynomials. Oxford University Press, Oxford (1995) · Zbl 0824.05059
[10]	Minahan J.A., Polychronakos A.P.: Density correlation functions in Calogero-Sutherland models. Phys. Rev. B 50, 4236-4239 (1994)
[11]	Nazarov, M.L., Sklyanin, E.K.: Macdonald operators at infinity. J. Algebr. Comb. (2013). doi:10.1007/s10801-013-0477-2 · Zbl 1297.05242
[12]	Okounkov A., Pandharipande R.: Quantum cohomology of the Hilbert scheme of points in the plane. Inv. Math. 179, 523-557 (2010) · Zbl 1198.14054 · doi:10.1007/s00222-009-0223-5
[13]	Pogrebkov A.K.: Boson-fermion correspondence and quantum integrable and dispersionless models. Russ. Math. Surv. 58, 1003-1037 (2003) · Zbl 1063.81065 · doi:10.1070/RM2003v058n05ABEH000668
[14]	Polychronakos A.P.: Waves and solitons in the continuum limit of the Calogero-Sutherland model. Phys. Rev. Lett. 74, 5153-5157 (1995) · Zbl 1020.81967 · doi:10.1103/PhysRevLett.74.5153
[15]	Rossi P.: Gromov -Witten invariants of target curves via Symplectic Field Theory. J. Geom. Phys. 58, 931-941 (2008) · Zbl 1145.53068 · doi:10.1016/j.geomphys.2008.02.012
[16]	Schiffmann, O., Vasserot, E.: Cherednik algebras, W-algebras and the equivariant cohomology of the moduli space of instantons on \[{\mathbb{A}^2}\] A2. Publ. Math. IHES (2013). http://link.springer.com/article/10.1007/s10240-013-0052-3 · Zbl 1284.14008
[17]	Sekiguchi J.: Zonal spherical functions on some symmetric spaces. Publ. Res. Inst. Math. Sci. 12, 455-459 (1977) · Zbl 0383.43005 · doi:10.2977/prims/1195196620
[18]	Shiraishi J.: A family of integral transformations and basic hypergeometric series. Comm. Math. Phys. 263, 439-460 (2006) · Zbl 1103.33011 · doi:10.1007/s00220-005-1504-5
[19]	Sklyanin E.K.: Separation of variables. New trends. Progress Theor. Phys. Suppl. 118, 35-60 (1995) · Zbl 0868.35002 · doi:10.1143/PTPS.118.35
[20]	Sutherland B.: Exact results for a quantum many-body problem in one dimension. Phys. Rev. A 4, 2019-2021 (1971)
[21]	Sutherland B.: Exact results for a quantum many-body problem in one dimension II. Phys. Rev. A 5, 1372-1376 (1972) · Zbl 0277.34049