The correlation functions of vertex operators and Macdonald polynomials
Shun-Jen Cheng1
and Weiqiang Wang2
1Academia Sinica Institute of Mathematics Taipei Taiwan 115 Taipei Taiwan 115
2University of Virginia Department of Mathematics Charlottesville VA USA 22904 Charlottesville VA USA 22904
2University of Virginia Department of Mathematics Charlottesville VA USA 22904 Charlottesville VA USA 22904
DOI: 10.1007/s10801-006-0022-7
Abstract
The n-point correlation functions introduced by Bloch and Okounkov have already found several geometric connections and algebraic generalizations. In this note we formulate a q, t-deformation of this n-point function. The key operator used in our formulation arises from the theory of Macdonald polynomials and affords a vertex operator interpretation. We obtain closed formulas for the n-point functions when n = 1,2 in terms of the basic hypergeometric functions. We further generalize the q, t-deformed n-point function to more general vertex operators.
Pages: 43–56
Keywords: keywords correlation functions; Macdonald polynomials; vertex operators; hypergeometric series
Full Text: PDF
References
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2. S.-J. Cheng and W. Wang, “The Bloch-Okounkov correlation functions at higher levels,” Transform. Groups 9(2) (2004), 133-142.
3. C. Dong and G. Mason, “Monstrous Moonshine of higher weight,” Acta Math. 185 (2000), 101-121.
4. A. Garsia and M. Haiman, “A remarkable q,t-Catalan sequence and q-Lagrange inversion,” J. Algebraic Combin. 5 (1996), 191-244.
5. G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd edition, Encyclopedia of Mathematics and its Applications vol.
96. Cambridge Univ. Press, Cambridge, 2004.
6. A. Lascoux and J.-Y. Thibon, “Vertex operators and the class algebras of symmetric groups,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 283 (2001), 156-177, math.CO/0102041.
7. W.-P. Li, Z. Qin, and W. Wang, “Hilbert schemes, integrable hierarchies, and Gromov-Witten theory,” Int. Math. Res. Not. 40 (2004), 2085-2104.
8. I.G. Macdonald, Symmetric Functions and Hall polynomials, 2nd edition, Oxford Mathematical Monographs. The Clarendon Press, Oxford Univ. Press, New York, 1995.
9. A. Milas, “Formal differential operators, vertex operator algebras and zeta-values, II,” J. Pure Appl. Al- gebra 183 (2003), 191-244.
10. A. Okounkov, “Infinite wedge and random partitions,” Selecta Math. (N.S.) 7 (2001), 1-25.
11. A. Okounkov and R. Pandharipande, Gromov-Witten Theory, Hurwitz Numbers, and Completed Cycles, Ann. Math. 163 (2006), 517-560.
12. W. Wang, “Correlation functions of strict partitions and twisted Fock spaces,” Transform. Groups 9(1) (2004), 89-101.