# zbMATH — the first resource for mathematics

Generalised discriminants, deformed Calogero-Moser-Sutherland operators and super-Jack polynomials. (English) Zbl 1069.05076
Summary: It is shown that the deformed Calogero-Moser-Sutherland (CMS) operators can be described as the restrictions on certain affine subvarieties (called generalised discriminants) of the usual CMS operators for infinite number of particles. The ideals of these varieties are shown to be generated by the Jack symmetric functions related to the Young diagrams with special geometry. A general structure of the ideals which are invariant under the action of the quantum CMS integrals is discussed in this context. The shifted super-Jack polynomials are introduced and combinatorial formulas for them and for super-Jack polynomials are given.

##### MSC:
 05E05 Symmetric functions and generalizations 81R12 Groups and algebras in quantum theory and relations with integrable systems 33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
Full Text:
##### References:
 [1] Calogero, F., Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. math. phys., 12, 419-436, (1971) [2] A. Cayley, Collected Mathematical Papers, Vol. II, Cambridge University Press, Cambridge, 1889 (reprinted). [3] Chalykh, O.A.; Feigin, M.V.; Veselov, A.P., New integrable generalizations of Calogero-Moser quantum problem, J. math. phys., 39, 2, 695-703, (1998) · Zbl 0906.34061 [4] Chipalkatti, J.V., On equations defining coincident root loci, J. algebra, 267, 246-271, (2003) · Zbl 1099.13501 [5] Feigin, B.; Jimbo, M.; Miwa, T.; Mukhin, E., A differential ideal of symmetric polynomials spanned by Jack polynomials at b=−(r−1)/(k+1), Internat. math. res. notices, 23, 1223-1237, (2002) · Zbl 1012.05153 [6] Feigin, M.; Veselov, A.P., Quasi-invariants and quantum integrals of the deformed Calogero-Moser systems, Internat. math. res. notices, 46, 2487-2511, (2003) · Zbl 1034.81024 [7] Gefand, I.M.; Kapranov, M.M.; Zelevinsky, A.V., Discriminants, resultants and multidimensional determinants, (1994), Birkhäuser Boston · Zbl 0827.14036 [8] Ivanov, V.; Olshanski, G., Kerov’s central limit theorem for the Plancherel measure on Young diagrams, () · Zbl 1016.05073 [9] M. Kasatani, T. Miwa, A.N. Sergeev, A.P. Veselov, Coincident root loci and Jack and Macdonald polynomials for special values of the parameters, math.QA/0404079. · Zbl 1188.33019 [10] Kerov, S.; Okounkov, A.; Olshanski, G., The boundary of the Young graph with Jack edge multipliers, Internat. math. res. notices, 4, 173-199, (1998) · Zbl 0960.05107 [11] Knop, F.; Sahi, S., Difference equations and symmetric polynomials defined by their zeros, Internat. math. res. notices, 10, 437-486, (1996) · Zbl 0880.43014 [12] Lapoint, L.; Lascoux, A.; Morse, J., Determinantal expression and recursion for Jack polynomials, Electron. J. combin., 7(1), 7, (2000), (Note 1) · Zbl 0934.05123 [13] Macdonald, I., Symmetric functions and Hall polynomials, (1995), Oxford University Press Oxford · Zbl 0824.05059 [14] Okounkov, A., (shifted) Macdonald polynomialsq-integral representation and combinatorial formula, Compositio math., 112, 2, 147-182, (1998) · Zbl 0897.05085 [15] A. Okounkov, On N-point correlations in the log-gas at rational temperature, hep-th/9702001. [16] Okounkov, A.; Olshanski, G., Shifted Jack polynomials, binomial formula, and applications, Math. res. lett., 4, 69-78, (1997) · Zbl 0995.33013 [17] E. Opdam, Lectures on Dunkl operators, math.RT/9812007. [18] A. Regev, On a class of algebras defined by partitions, math.CO/0210195. · Zbl 1039.16025 [19] Sahi, S., The spectrum of certain invariant differential operators associated to a Hermitian symmetric space, Prog. math., 123, 569-576, (1994) · Zbl 0851.22010 [20] Sergeev, A.N., Superanalogs of the Calogero operators and Jack polynomials, J. nonlinear math. phys., 8, 1, 59-64, (2001) · Zbl 0972.35114 [21] Sergeev, A.N., Calogero operator and Lie superalgebras, Theoret. math. phys., 131, 3, 747-764, (2002) · Zbl 1039.81028 [22] Sergeev, A.N.; Veselov, A.P., Deformed quantum Calogero-Moser systems and Lie superalgebras, Comm. math. phys., 245, 2, 249-278, (2004) · Zbl 1062.81097 [23] Stanley, R., Some combinatorial properties of Jack symmetric functions, Adv. math., 77, 76-115, (1989) · Zbl 0743.05072 [24] Sutherland, B., Exact results for a quantum many-body problem in one dimension, Phys. rev. A, 4, 2019-2021, (1971) [25] Vershik, A.M.; Kerov, S.V., Asymptotic theory of characters of the symmetric group, Funct. anal. appl., 19, 21-31, (1985) · Zbl 0592.20015 [26] Veselov, A.P.; Feigin, M.V.; Chalykh, O.A., New integrable deformations of quantum Calogero-Moser problem, Russian math. surveys, 51, 3, 185-186, (1996) · Zbl 0874.35098 [27] Weyman, J., The equations of strata for binary forms, J. algebra, 122, 244-249, (1989) · Zbl 0689.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.