Jack polynomials and some identities for partitions

Lassalle, Michel

doi:10.1090/S0002-9947-04-03500-7

Jack polynomials and some identities for partitions
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Trans. Amer. Math. Soc. 356 (2004), 3455-3476 Request permission

Abstract:

We prove an identity about partitions involving new combinatorial coefficients. The proof given is using a generating function. As an application we obtain the explicit expression of two shifted symmetric functions, related with Jack polynomials. These quantities are the moments of the “$\alpha$-content” random variable with respect to some transition probability distributions.

References

Frédéric Jouhet and Jiang Zeng, Généralisation de formules de type Waring, Sém. Lothar. Combin. 44 (2000), Art. B44g, 9 (French, with English summary). MR 1814860
F. Jouhet, B. Lass, Jiang Zeng, Sur une généralisation des coefficients binomiaux, http:// arXiv.org/abs/math.CO/0303025.
Jyoichi Kaneko, Selberg integrals and hypergeometric functions associated with Jack polynomials, SIAM J. Math. Anal. 24 (1993), no. 4, 1086–1110. MR 1226865, DOI 10.1137/0524064
S. V. Kerov, Anisotropic Young diagrams and symmetric Jack functions, Funktsional. Anal. i Prilozhen. 34 (2000), no. 1, 51–64, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 34 (2000), no. 1, 41–51. MR 1756734, DOI 10.1007/BF02467066
S. Kerov, The boundary of Young lattice and random Young tableaux, Formal power series and algebraic combinatorics (New Brunswick, NJ, 1994) DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 24, Amer. Math. Soc., Providence, RI, 1996, pp. 133–158. MR 1363510, DOI 10.1007/bf02362775
Sergei Kerov, Andrei Okounkov, and Grigori Olshanski, The boundary of the Young graph with Jack edge multiplicities, Internat. Math. Res. Notices 4 (1998), 173–199. MR 1609628, DOI 10.1155/S1073792898000154
Friedrich Knop, Symmetric and non-symmetric quantum Capelli polynomials, Comment. Math. Helv. 72 (1997), no. 1, 84–100. MR 1456318, DOI 10.4171/CMH/72.1.7
Friedrich Knop and Siddhartha Sahi, Difference equations and symmetric polynomials defined by their zeros, Internat. Math. Res. Notices 10 (1996), 473–486. MR 1399412, DOI 10.1155/S1073792896000311
A. Lascoux, Notes on interpolation in one and several variables, http://phalanstere.univ-mlv.fr/~al/.
Alain Lascoux and Michel Lassalle, Une identité remarquable en théorie des partitions, Math. Ann. 318 (2000), no. 2, 299–313 (French, with English summary). MR 1795564, DOI 10.1007/s002080000121
M. Lassalle, A new family of positive integers, Ann. Comb. 6 (2002), 399–405.
Michel Lassalle, Une identité en théorie des partitions, J. Combin. Theory Ser. A 89 (2000), no. 2, 270–288 (French, with English summary). MR 1741013, DOI 10.1006/jcta.1999.3010
Michel Lassalle, Quelques conjectures combinatoires relatives à la formule classique de Chu-Vandermonde, Adv. in Appl. Math. 21 (1998), no. 3, 457–472 (French). MR 1641230, DOI 10.1006/aama.1998.0606
Michel Lassalle, Some combinatorial conjectures for Jack polynomials, Ann. Comb. 2 (1998), no. 1, 61–83. MR 1682920, DOI 10.1007/BF01626029
Michel Lassalle, Some combinatorial conjectures for shifted Jack polynomials, Ann. Comb. 2 (1998), no. 2, 145–163. MR 1682926, DOI 10.1007/BF01608485
Michel Lassalle, Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal. 158 (1998), no. 2, 289–324 (French). MR 1648471, DOI 10.1006/jfan.1998.3281
Michel Lassalle, Une formule du binôme généralisée pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), no. 5, 253–256 (French, with English summary). MR 1042857
Michel Lassalle, Une formule de Pieri pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 18, 941–944 (French, with English summary). MR 1054739
I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
G. Olshanski, A. Regev, Random Young tableaux and combinatorial identities, http:// arXiv.org/abs/math.CO/0106074.
Andrei Okounkov, (Shifted) Macdonald polynomials: $q$-integral representation and combinatorial formula, Compositio Math. 112 (1998), no. 2, 147–182. MR 1626029, DOI 10.1023/A:1000436921311
A. Okounkov and G. Olshanski, Shifted Jack polynomials, binomial formula, and applications, Math. Res. Lett. 4 (1997), no. 1, 69–78. MR 1432811, DOI 10.4310/MRL.1997.v4.n1.a7
A. Okun′kov and G. Ol′shanskiĭ, Shifted Schur functions, Algebra i Analiz 9 (1997), no. 2, 73–146 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 9 (1998), no. 2, 239–300. MR 1468548
Anatol N. Kirillov and Masatoshi Noumi, Affine Hecke algebras and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998), no. 1, 1–39. MR 1620075, DOI 10.1215/S0012-7094-98-09301-2
Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073, DOI 10.1016/0001-8708(89)90015-7
S. Veigneau, ACE, an Algebraic Combinatorics Environment for the computer algebra system Maple, http://phalanstere.univ-mlv.fr/~ace/.
Jiang Zeng, private communication.