×

Representation-theoretic proof of the inner product and symmetry identities for Macdonald’s polynomials. (English) Zbl 0859.17005

In 1988 I. G. Macdonald introduced a new class of symmetric polynomials in Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Séminaire Lotharingien, 131-171 (1988). In a previous paper the authors of the work under review obtained those Macdonald’s polynomials related to the root system \(A_{n-1}\) via finite-dimensional representation of the quantum group \(U_q{\mathfrak{sl}}_n\). Here they give a representation-theoretic proof of Macdonald’s inner product and symmetry identities in the \(A_{n-1}\)-case. Previously, analogous results had been obtained by combinatorial methods by Macdonald and, in the more general case, by Cherednik. The proof uses intertwining operators and Shapovalov’s formula as well as the ribbon graphs of Reshetikhin and Turaev.

MSC:

17B37 Quantum groups (quantized enveloping algebras) and related deformations
05E05 Symmetric functions and generalizations
PDFBibTeX XMLCite
Full Text: arXiv Numdam EuDML

References:

[1] Askey, R. and Ismail, Mourad E.-H. : A generalization of ultraspherical polynomials, Studies in Pure Math . (P. Erdös, ed.), Birkhäuser, 1983, pp. 55-78. · Zbl 0532.33006
[2] Cherednik, I. : Double affine Hecke algebras and Macdonald’s conjectures , Annals of Math. 141 (1995) 191-216. · Zbl 0822.33008 · doi:10.2307/2118632
[3] Cherednik, I. : Macdonald’s evaluation conjectures and difference Fourier transform , preprint, December 1994, q-alg/9412016. · Zbl 0854.22021 · doi:10.1007/BF01231441
[4] De Concini, C. and Kac, V.G. : Representations of quantum groups at roots of 1, Operator algebras, Unitary Representations, Enveloping Algebras and Invariant Theory (A. Connes et al, eds.), Birkhäuser, 1990, pp. 471-506. · Zbl 0738.17008
[5] Drinfeld, V.G. : Quantum groups, Proc. Int. Congr. Math. , Berkeley, 1986, pp. 798-820 · Zbl 0667.16003
[6] Etingof, P.I. and Kirillov, A.A., Jr. : On a unified representation-theoretic approach to the theory of special functions , Funktsion. analiz i ego prilozh. 28 (1994) no. 1, 91-94 (in Russian). · Zbl 0868.33010 · doi:10.1007/BF01079011
[7] Etingof, P.I. and Kirillov, A.A., Jr. : Macdonald’s polynomials and representations of quantum groups , Math. Res. Let. 1 (1994) no. 3, 279-296. · Zbl 0833.17007 · doi:10.4310/MRL.1994.v1.n3.a1
[8] Etingof, P.I. and Styrkas, K. : Algebraic integrability of Schrödinger operators and representations of Lie algebras , preprint, hep-th/9403135 (1994), to appear in Compositio Math. · Zbl 0861.17003
[9] Jimbo, M.A. : A q-difference analogue of Ug and the Yang-Baxter equation , Lett. Math. Phys. 10 (1985) 62-69. · Zbl 0587.17004 · doi:10.1007/BF00704588
[10] Lusztig, G. : Introduction to quantum groups , Birkhäuser, Boston, 1993. · Zbl 0788.17010
[11] Macdonald, I.G. : A new class of symmetric functions , Publ. I.R.M.A. Strasbourg, 372/S-20, Actes 20 Séminaire Lotharingien (1988), 131-171. · Zbl 0962.05507
[12] Macdonald, I.G. : Orthogonal polynomials associated with root systems , preprint (1988). · Zbl 0699.42010
[13] Reshetikhin, N. and Turaev, V. : Ribbon graphs and their invariants derived from quantum groups , Comm. Math. Phys. 127 (1990) 1-26. · Zbl 0768.57003 · doi:10.1007/BF02096491
[14] Reshetikhin, N. and Turaev, V. : Invariants of 3-manifolds via link polynomials and quantum groups , Inv. Math. 103 (1991) 547-597. · Zbl 0725.57007 · doi:10.1007/BF01239527
[15] Tanisaki, T. : Killing forms, Harish-Chandra isomorphisms and universal R-matrices for quantum algebras, Infinite Analysis, part A and part B (Kyoto, 1991), Adv. Ser. Math. Phys. 17, World Scientific, pp. 941-961. · Zbl 0870.17007 · doi:10.1142/S0217751X92004117
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.