Symmetric functions, codes of partitions and the KP hierarchy
S.R. Carrell
and I.P. Goulden2
2S.R. Carrell
DOI: 10.1007/s10801-009-0211-2
Abstract
We consider an operator of Bernstein for symmetric functions and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ -function for the hierarchy. This proof is especially compact because we are able to restate the Plücker relations in a form that is symmetrical in terms of partition code notation.
Pages: 211–226
Keywords: symmetric functions; Schur functions; plücker relations; KP hierarchy; combinatorial bijection; partition code
Full Text: PDF
References
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2. Goulden, I.P., Jackson, D.M.: The KP hierarchy, branched covers, and triangulations. Adv. Math. 219, 932-951 (2008)
3. Hoffman, P.N.: Littlewood-Richardson without algorithmically defined bijections. Séminaire Lotharingien de Combinatoire, B20k (1988), 6 pp
4. Hoffman, P.N.: A Bernstein-type formula for projective representations of Sn and An. Adv. Math. 74, 135-143 (1989)
5. Hoffman, P.N., Humphreys, J.F.: Projective Representations of the Symmetric Groups. Q-Functions and Shifted Tableaux. Oxford Mathematical Monographs. Oxford Science Publications. Clarendon/Oxford University Press, New York (1992), xiv+304 pp
6. Jarvis, P.D., Yung, C.M.: Symmetric functions and the KP and BKP hierarchies. J. Phys. A 26, 5905- 5922 (1993)
7. Kazarian, M.: KP hierarchy for Hodge integrals. Adv. Math. 221, 1-21 (2009)
8. Kazarian, M., Lando, S.: An algebro-geometric proof of Witten's conjecture. J. Am. Math. Soc. 20, 1079-1089 (2007)
9. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147, 1-23 (1992)
10. Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, London (1995)
11. Miwa, T., Jimbo, M., Date, E.: Solitons: Differential Equations, Symmetries and Infinite Dimensional Algebras. Cambridge University Press, Cambridge (2000)
12. Okounkov, A.: Toda equations for Hurwitz numbers. Math. Res. Lett. 7, 447-453 (2000)
13. Pandharipande, R.: The Toda equations and the Gromov-Witten theory of the Riemann sphere. Lett.
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