Anisotropic Young Diagrams and Jack Symmetric Functions

S. V. Kerov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 34 (2000) no. 1, pp. 51-64.

Résumé

We study the Young lattice with the edge multiplicities $\varkappa_\alpha(\lambda,\Lambda)$ arising in the simplest Pieri formula for Jack symmetric polynomials $P_\lambda(x;\alpha)$ with parameter $\alpha$. A new proof of Stanley's $\alpha$-version of the hook formula is given. We also prove the formula $$ \sum_\Lambda (c_\alpha(b)+u)(c_\alpha(b)+v)\varkappa_\alpha(\lambda,\Lambda)\varphi(\Lambda)= (n\alpha+uv)\varphi(\lambda), $$ where $\varphi(\lambda)=\prod_{b\in\lambda}(a(b)\alpha+l(b)+1)^{-1}$ and $c_\alpha(b)$ is the $\alpha$-contents of the new box $b=\Lambda\setminus\lambda$.

Bibliographie

[1] Kerov S. V., “A $q$-analog of the hook walk algorithm for random Young tableaux”, J. Algebraic Combin., 2 (1993), 383–396 | DOI | MR | Zbl

[2] Kerov S. V., “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., 24, Amer. Math. Soc., Providence, RI, 1996, 133–158 | DOI | MR | Zbl

[3] Kerov S. V., “Asimptotika razdeleniya kornei ortogonalnykh mnogochlenov”, Algebra i analiz, 5:5 (1993), 68–86 | MR

[4] Kerov S. V., “Perekhodnye veroyatnosti kontinualnykh diagramm Yunga i problema momentov Markova”, Funkts. analiz i ego pril., 27:2 (1993), 32–49 | MR | Zbl

[5] Kerov S. V., “Interlacing measures”, Kirillov's seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, 181, Amer. Math. Soc., Providence, RI, 1998, 35–83 | MR | Zbl

[6] Kerov S. V., Olshanski G. I., Vershik A. M., “Harmonic analysis on the infinite symmetric group”, C. R. Acad. Sci. Paris Sér. I Math., 316 (1993), 773–778 | MR | Zbl

[7] Kerov S. V., Vershik A. M., “The Grothendieck group of the infinite symmetric group and symmetric functions with the elements of the $K_0$-functor theory of AF-algebras”, Representation of Lie groups and related topics, Adv. Stud. Contemp. Math., 7, eds. A. M. Vershik and D. P. Zhelobenko, Gordon and Breach, 1990, 36–114 | MR

[8] Kirillov A. N., “Tozhdestvo Lagranzha i formula kryukov”, Zap. semin. LOMI, 172, 1989, 78–87 | MR

[9] Macdonald I. G., Symmetric functions and Hall polynomials, 2nd edition, Oxford University Press, 1995 | MR | Zbl

[10] Stanley R. P., “Some combinatorial properties of Jack symmetric functions”, Adv. Math., 77 (1989), 76–115 | DOI | MR | Zbl

[11] Vershik A. M., “Formula kryukov i svyazannye s nei tozhdestva”, Zap. semin. LOMI, 172, 1989, 3–20 | MR

[12] Vershik A. M., Kerov S. V., “Asimptotika maksimalnoi i tipichnoi razmernostei neprivodimykh predstavlenii simmetricheskoi gruppy”, Funkts. analiz i ego pril., 19:1 (1985), 25–36 | MR | Zbl

[13] Kerov S. V., Okounkov A. and Olshanski G. I., “The boundary of Young graph with Jack edge multiplicities”, Internat. Math. Res. Notices, 4 (1998), 173–199 ; arXiv: /q-alg/9703037 | DOI | MR | Zbl

[14] Frame J. S., Robinson G. de B., Thrall R. M., “The hook graphs of the symmetric group”, Canad. J. Math., 6 (1954), 316–324 | DOI | MR | Zbl

[15] Rutherford D. E., “On the relations between the numbers of standard tableaux”, Proc. Edinburgh Math. Soc., 7 (1942–46), 51–54 | DOI | MR | Zbl

[16] Shensted C., “Longest increasing and decreasing subsequences”, Canad. J. Math., 13 (1961), 179–191 | DOI | MR