Annals of Combinatorics 4 (2000) 383-400
Skew Schur Functions and Yangian Actions on Irreducible Integrable Modules of
Research Institute for Mathematical Sciences, Kyoto University, 606
Received March 26, 1999
AMS Subject Classification: 17B67, 17B37, 17B81
Abstract. An action of the Yangian of the general linear Lie algebra is defined on every irreducible integrable highest weight module of . This action is derived, by means of the Drinfeld duality and a subsequent semi-infinite limit, from a certain induced representation of the degenerate double affine Hecke algebra Each vacuum modul e of is decomposed into irreducible Yangian representations by means of the intertwiners of Components of this decomposition are parameterized by semi-infinite skew Young diagrams. The decomposition gives rise to a character formulas for the modules of in terms of skew Schur functions.
Keywords: Representation of affine Lie algebra, Yangian action , skew Young diagram, skew schur functions
1. T. Arakawa, T. Nakanishi, K. Oshima, and A. Tsuchiya, Spectral decomposition of path space in solvable lattice models, Comm. Math. Phys. 181 (1996) 159–182.
2. T. Arakawa, T. Suzuki, and A. Tsuchiya, Degenerate double affine Hecke algebra and CFT, RIMS-1186, q-alg/9710031, 1997, preprint.
3. D. Bernard, V. Pasquier, and D. Serban, Spinons in conformal field theory, Nucl. Phys. B428 (1994) 612–628.
4. P. Bouwknegt, A. Ludwig, and K. Schoutens, Spinon bases, Yangian symmetry and fermionic representations of Virasoro characters in conformal field theory, Phys. Lett. 338B (1994) 448–456.
5. I. Cherednik, A new interpretation of Gelfand–Zetlin bases, Duke Math. J. 54 (1987) 563– 577.
6. I. Cherednik, Induced representations of double affine Hecke algebras and application, Math. Res. Lett. 1 (1994) 319–337.
7. I. Cherednik, Elliptic quantum many-body problem and double affine Knizhnik– Zamolodchikov equation, Comm. Math. Phys. 169 (1995) 441–461.
8. E. Date, M. Jimbo, A. Kuniba, T. Miwa, and M. Okado, Paths, maya diagrams and representations of , Adv. Stud. in Pure Math. 19 (1989) 149–191.
9. I. Frenkel, Representations of affine Lie algebras, Hecke modular form and Korteweg–de Vries type equation, in: Lie Algebra and Related Topics, D. Winter, Ed., Lect Notes in Mathmatics, Vol. 933, Spring-Verlag, 1982, pp. 71–110.
10. V.G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986) 62–64.
11. V.G. Drinfeld, A new realization of Yangians and quantized affine algebras, Sov. Math. Dokl. 36 (1988) 212–216.
12. F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, D. Bernard, and V. Pasquier, Yangian symmetry of integrable quantum chains with long-range interactions and a new description of states in conformal field theory, Phys. Rev. Lett. 69 (1992) 2012–2025.
13. A. Kirillov, A. Kuniba and T. Nakanishi, Skew Young diagram method in spectral decomposition of integrable lattice models, Comm. Math. Phys. 185 (1997) 441–465.
14. A. Kirillov, A. Kuniba, and T. Nakanishi, Skew Young diagram method in spectral decomposition of integrable lattice models II: Higher levels, q-alg/971109, 1997, preprint.
15. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Clarendon Press, Oxford, 1979.
16. M. Nazarov, and V. Tarasov, Representations of Yangians with Gelfand–Zetlin bases, J. Reine Angew. Math. 496 (1998) 181–212.
17. K. Schoutens, Yangian symmetry in conformal field theory, Phys. Lett. 331B (1994) 335– 341.
18. D. Uglov, Symmetric functions and the Yangian decompositions of Fock and basic modules of , Memoires of the Mathematical Society of Japan, 1 (1998) 183–241.
19. D. Uglov, Yangian actions on higher level irreducible integrable modules of , RIMS- 1184, math. QA/9802048, 1998, preprint.