On vertex operator realizations of Jack functions
Wuxing Cai
and Naihuan Jing
DOI: 10.1007/s10801-010-0228-6
Abstract
On the vertex operator algebra associated with a rank one lattice we derive a general formula for products of vertex operators in terms of generalized homogeneous symmetric functions. As an application we realize Jack symmetric functions of rectangular shapes as well as marked rectangular shapes.
Pages: 579–595
Keywords: keywords symmetric functions; Jack polynomials; vertex operators
Full Text: PDF
References
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2. Garsia, A.: Orthogonality of Milne's polynomials and raising operators. Discrete Math. 99(1-3), 247- 264 (1992)
3. Jack, H.: A class of symmetric polynomials with a parameter. Proc. R. Soc. Edinb. Sect. A 69, 1-18 (1970/1971)
4. Jing, N.: Vertex operators, Vertex operators, symmetric functions, and the spin group Γn. J. Algebra 138(2), 340-398 (1991)
5. Jing, N.: Vertex operators and Hall-Littlewood symmetric functions. Adv. Math. 87(2), 226-248 (1991)
6. Jing, N.: q-hypergeometric series and Macdonald functions. J. Algebr. Comb. 3, 291-305 (1994)
7. Jing, N., Józefiak, T.: A formula for two row Macdonald functions. Duke Math. J. 67(2), 377-385 (1992) (1)
8. Lepowsky, J., Primc, M.: Structure of the Standard Modules for the Affine Lie Algebra A . Cont. 1 Math., vol.
46. Am. Math. Soc., Providence (1985)
9. Lepowsky, J., Wilson, R.: The structure of standard modules. I. Universal algebras and the Rogers- Ramanujan identities. Invent. Math. 77(2), 199-290 (1984)
10. Macdonald, I.G.: Symmetric Unctions and Hall Polynomials, 2nd edn. With Contributions by A. Zelevinsky. Oxford Univ. Press, New York (1995)
11. Mimachi, K., Yamada, Y.: Singular vectors of the Virasoro algebra in terms of Jack symmetric poly- nomials. Commun. Math. Phys. 174(2), 447-455 (1995)
12. Wakimoto, M., Yamada, H.: Irreducible decompositions of representations of the Virasoro algebra. Lett. Math. Phys. 7(6), 513-516 (1983)
13. Stanley, R.: Some combinatorial properties of Jack symmetric functions. Adv. Math. 77(1), 76-115 (1989)
14. Zabrocki, M.: A Macdonald vertex operator and standard tableaux statistics for the two-column (q, t)- Kostka coefficients. Electron. J. Comb. 5, 45 (1998)
15. Zelevinsky, A.: Representations of Finite Classical Groups. Lecture Notes in Math., vol. 869.
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