Abstract
We consider some remarkable central elements of the universal enveloping algebraU(gl(n)) which we call quantum immanants. We express them in terms of generatorsE ij ofU(gl(n)) and as differential operators on the space of matrices These expressions are a direct generalization of the classical Capelli identities. They result in many nontrivial properties of quantum immanants.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[BL1] L. C. Biedenharn and J. D. Louck,A new class of symmetric polynomials defined in terms of tableaux, Advances in Appl. Math.10 (1989), 396–438.
[BL2]—Inhomogeneous basis set of symmetric polynomials defined by tableaux, Proc. Nat. Acad. Sci. U.S.A.87 (1990), 1441–1445.
[C] A. Capelli,Über die Zurückführung der Cayley'schen Operation Ω auf gewöhnlichen Polar-Operationen, Math. Ann.29 (1887), 331–338.
[GG] I. Goulden and C. Greene,A new tableau representation for supersymmetric Schur functions, J. Algebra170 (1994), 687–704.
[Ch] I. V. Cherednik,On special bases of irreducible finite-dimensional representations of the degenerated affine Hecke algebra, Funct. Anal. Appl.20 (1986), no. 1, 87–89.
[GH] I. P. Goulden and A. M. Hamel,Shift operators and factorial symmetric functions, University of Waterloo, J. Comb. Theor. A.69 (1995), 51–60.
[CL] W. Y. C. Chen and J. D. Louck,The factorial Schur function, J. Math. Phys.34 (1993), 4144–4160.
[H] R. Howe,Remarks on classical invariant theory, Trans. AMS313 (1989), 539–570.
[HU] R. Howe and T. Umeda,The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann.290 (1991), 569–619.
[JK] G. James and A. Kerber,The representation theory of the symmetric group. Encyclopedia of mathematics and its applications, vol. 16, Addison-Wesley, 1981.
[JKMO] M. Jimbo, A. Kuniba, T. Miwa and M. Okado,The A (1)n face models, Commun. Math. Phys.119 (1988), 543–565.
[KO] S. Kerov and G. Olshanski,Polynomial functions on the set of Young diagrams, Comptes Rendus Acad. Sci. Paris, Sér. I319 (1994), 121–126.
[KS1] B. Kostant and S. Sahi,The Capelli, identity, tube domains and the generalized Laplace transform, Advances in Math.87 (1991), 71–92.
[KS2]—,Jordan algebras and Capelli identities, Invent. Math.112 (1993), 657–664.
[KuR] P. P. Kulish and N. Yu. Reshetikhin,GL 3-invariant solutions of the Yang-Baxter equation, J. Soviet Math.34 (1986), 1948–1971.
[KuRS] P. P. Kulish, N. Yu. Reshetikhin and E. K. Sklyanin,Yang-Baxter equation end representation theory, Lett. Math. Phys.5 (1981), 393–403.
[KuS] P. P. Kulish and E. K. Sklyanin,Quantum spectral transform method: recent developments, Integrable Quantum Field Theories, Lecture Notes in Phys., vol. 151, Springer Verlag, Berlin-Heidelberg, 1982, pp. 61–119.
[M1] I. G. Macdonald,Symmetric functions and Hall polynomials, Oxford University Press, 1979.
[M2] I. G. Macdonald,Schur functions: theme and variations, Publ. I.R.M.A. Strasbourg 498/S-27 Actes 28-e Séminaire Lotharingien (1992), 5-39.
[MNO] A. Molev, M. Nazarov and G. Olshanski,Yangians and classical Lie algebras, to appear in Russ. Math. Surv., Australian Nat. Univ. Research Report (1993), 1–105.
[N] M. L. Nazarov,Quantum Berezinian and the classical Capelli identity, Letters in Math. Phys.21 (1991), 123–131.
[NUW] M. Noumi, T. Umeda and M. Wakayama,A quantum analogue of the Capelli identity and an elementary differential calculus on GL q(n) Duke Math. J.76 (1994), no. 2.
[Ol1] G. Olshanski,Representations of infinite-dimensional classical groups, limits of enveloping algebras, and Yangians, Topics in representation theory, Advances in Soviet Mathematics (A. Kirillov, ed.), vol. 2, AMS, Providence, RI, 1991, pp. 1–66.
[Ol2] G. Olshanski,Quasi-symmetric functions and factorial Schur functions, preprint (1995).
[OO] A. Okounkov and G. Olshanski,Shifted Schur functions, to appear.
[RTF] N. Reshetikhin, L. Takhtajan and L. Faddeev,Quantization of Lie Groups and Lie algebras, Leningrad Math. J.1 (1990), 193–225.
[S] S. Sahi,The spectrum of certain invariant differential operators associated to a Hermitian symmetric space, Lie Theory and Geometry: in Honour of Bertram Kostant, Progress in Mathematics (J.-L. Brylinski, R. Brylinski, V. Guillemin, V. Kac, eds.), vol. 123, Birkhäuser, Boston, Basel, 1994.
[VK1] A. Vershik and S. Kerov,Asymptotic theory of characters of the infinite symmetric group, Funct. Anal. Appl.15 (1981), 246–255.
[VK2]—,Characters and factor representations of the infinite unitary group, Soviet Math. Dokl.26 (1982), 570–574.
Author information
Authors and Affiliations
Additional information
The author is supported by the International Science Foundation and the Russian Fundamental Research Foundation.
Rights and permissions
About this article
Cite this article
Okounkov, A. Quantum immanants and higher Capelli identities. Transformation Groups 1, 99–126 (1996). https://doi.org/10.1007/BF02587738
Received:
Revised:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02587738