The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications

Han, Guo-Niu

Annales de l'Institut Fourier, Tome 60 (2010) no. 1, pp. 1-29.

Nekrasov et Okounkov ont obtenu une nouvelle formule pour le développement des puissances du produit d’Euler, à l’aide des longueurs d’équerre des partitions d’entiers, dans leur étude de la théorie de Seiberg-Witten. Nous proposons un raffinement de cette formule reposant sur une propriété nouvelle des $t$-cores, qui permet de donner une démonstration élémentaire en faisant usage des identités de Macdonald. Nous obtenons aussi une extension, en ajoutant deux paramètres supplémentaires, qui peut être considérée comme une interpolation discrète entre les identités de Macdonald et la fonction génératrice des $t$-cores. Plusieurs applications en sont déduites, y compris la “formule d’équerre pointée”.

The paper is devoted to the derivation of the expansion formula for the powers of the Euler Product in terms of partition hook lengths, discovered by Nekrasov and Okounkov in their study of the Seiberg-Witten Theory. We provide a refinement based on a new property of $t$-cores, and give an elementary proof by using the Macdonald identities. We also obtain an extension by adding two more parameters, which appears to be a discrete interpolation between the Macdonald identities and the generating function for $t$-cores. Several applications are derived, including the “marked hook formula”.

[1] Adin, R. M.; Frumkin, A. Rim Hook Tableaux and Kostant’s $\eta$-Function Coefficients, Adv. in Appl. Math., Volume 33 (2004), pp. 492-511 | DOI | MR | Zbl

[2] Andrews, G. E. The Theory of Partitions, Encyclopedia of Math. and Its Appl., 2, Addison-Wesley, Reading, 1976 | MR | Zbl

[3] Bacher, R.; Manivel, L. Hooks and Powers of Parts in Partitions, Sém. Lothar. Combin., Volume 47, 2001 (article B47d, 11 pages) | MR | Zbl

[4] Berkovich, A.; Garvan, F. G. The BG-rank of a partition and its applications, Adv. in Appl. Math., Volume 40 (2008), pp. 377-400 | DOI | MR

[5] Bessenrodt, C. On hooks of Young diagrams, Ann. of Comb., Volume 2 (1998), pp. 103-110 | DOI | MR | Zbl

[6] Carlsson, E.; Okounkov, A. Exts and Vertex Operators, arXiv:0801. 2565v1 [math.AG]

[7] Cellini, P.; Frajria, P. M.; Papi, P. The $\widehat{W}$-orbit of $\rho$, Kostant’s formula for powers of the Euler product and affine Weyl groups as permutations of $\mathbb{Z}$, J. Pure Appl. Algebra, Volume 208 (2007), pp. 1103-1119 | DOI | MR | Zbl

[8] Dyson, F. J. Missed opportunities, Bull. Amer. Math. Soc., Volume 78 (1972), pp. 635-652 | DOI | MR | Zbl

[9] Euler, L. The expansion of the infinite product $(1-x)(1-xx)(1-x^3)(1-x^4)(1-x^5)(1-x^6)$ etc. into a single series, English translation from the Latin by Jordan Bell (on arXiv:math.HO/0411454)

[10] Farkas, H. M.; Kra, I. On the Quintuple Product Identity, Proc. Amer. Math. Soc., Volume 27 (1999), pp. 771-778 | DOI | MR | Zbl

[11] Foata, D.; Han, G.-N. The triple, quintuple and septuple product identities revisited, Sem. Lothar. Combin. (Art. B42o, 12 pp) | Zbl

[12] Frame, J. S.; de Beauregard Robinson, G.; Thrall, R. M. The hook graphs of the symmetric groups, Canadian J. Math., Volume 6 (1954), pp. 316-324 | DOI | MR | Zbl

[13] Garvan, F.; Kim, D.; Stanton, D. Cranks and $t$-cores, Invent. Math., Volume 101 (1990), pp. 1-17 | DOI | MR | Zbl

[14] Gessel, I.; Viennot, G. Binomial determinants, paths, and hook length formulae, Adv. in Math., Volume 58 (1985), pp. 300-321 | DOI | MR | Zbl

[15] Greene, C.; Nijenhuis, A.; Wilf, H. S. A probabilistic proof of a formula for the number of Young tableaux of a given shape, Adv. in Math., Volume 31 (1979), pp. 104-109 | DOI | MR | Zbl

[16] Han, G.-N. An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths, arXiv:0804.1849v2, Math.CO, 2008 (35 pages)

[17] Han, G.-N. Discovering hook length formulas by an expansion technique, Electron. J. Combin., vol. 15(1), 2008 (Research Paper #R133, 41 pp) | MR | Zbl

[18] Hoare, A.; Howard, M. An Involution of Blocks in the Partitions of $n$, Amer. Math. Monthly, Volume 93 (1986), pp. 475-476 | DOI | MR | Zbl

[19] James, G.; Kerber, A. The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, 16, Addison-Wesley Publishing, Reading, MA, 1981 | MR | Zbl

[20] Joichi, J. T.; Stanton, D. An involution for Jacobi’s identity, Discrete Math., Volume 73 (1989), pp. 261-271 | DOI | MR | Zbl

[21] Kac, V. G. Infinite-dimensional Lie algebras and Dedekind’s $\eta$-function, Functional Anal. Appl., Volume 8 (1974), pp. 68-70 | DOI | MR | Zbl

[22] Kirdar, M. S.; Skyrme, T. H. R. On an Identity Related to Partitions and Repetitions of Parts, Canad. J. Math., Volume 34 (1982), pp. 194-195 | DOI | MR | Zbl

[23] Knuth, D. E. The Art of Computer Programming, Sorting and Searching, 2nd ed., 3, Addison Wesley Longman, 1998 | MR

[24] Kostant, B. On Macdonald’s $\eta$-function formula, the Laplacian and generalized exponents, Adv. in Math., Volume 20 (1976), pp. 179-212 | DOI | MR | Zbl

[25] Kostant, B. Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra, Invent. Math., Volume 158 (2004), pp. 181-226 | DOI | MR | Zbl

[26] Krattenthaler, C. Another involution principle-free bijective proof of Stanley’s hook-content formula, J. Combin. Theory Ser. A, Volume 88 (1999), pp. 66-92 | DOI | MR | Zbl

[27] Lascoux, Alain Symmetric functions and combinatorial operators on polynomials, CBMS Regional Conference Series in Mathematics, 99, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2003 | MR | Zbl

[28] Macdonald, I. G. Affine root systems and Dedekind’s $\eta$-function, Invent. Math., Volume 15 (1972), pp. 91-143 | DOI | MR | Zbl

[29] Macdonald, I. G. Symmetric Functions and Hall Polynomials, Second Edition, Clarendon Press, Oxford, 1995 | MR | Zbl

[30] Milne, S. C. An elementary proof of the Macdonald identities for $A_l^{\left(1\right)}$, Adv. in Math., Volume 57 (1985), pp. 34-70 | DOI | MR | Zbl

[31] Moody, R. V. Macdonald identities and Euclidean Lie algebras, Proc. Amer. Math. Soc., Volume 48 (1975), pp. 43-52 | DOI | MR | Zbl

[32] Nekrasov, N. A.; Okounkov, A. Seiberg-Witten theory and random partitions. The unity of mathematics, 244, Progr. Math., Birkhaeuser Boston (2006), pp. 525-596 (See also arXiv:hep-th/0306238v2, 90 pages, 2003) | MR

[33] Novelli, J.-C.; Pak, I.; Stoyanovskii, A. V. A direct bijective proof of the hook-length formula, Discrete Math. Theor. Comput. Sci., Volume 1 (1997), pp. 53-67 | MR | Zbl

[34] Remmel, J. B.; Whitney, R. A bijective proof of the hook formula for the number of column strict tableaux with bounded entries, European J. Combin., Volume 4 (1983), pp. 45-63 | MR | Zbl

[35] Rosengren, H.; Schlosser, M. Elliptic determinant evaluations and the Macdonald identities for affine root systems, Compositio Math., Volume 142 (2006), pp. 937-961 | DOI | MR | Zbl

[36] Serre, J.-P. Cours d’arithmétique, Collection SUP: “Le Mathématicien”, 2 Presses Universitaires de France, Paris, 1970 | MR | Zbl

[37] Sloane, N.; al. The On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/ | Zbl

[38] Stanley, R. P. Errata and Addenda to Enumerative Combinatorics Volume 1, Second Printing, version of 25 April 2008 http://www-math.mit.edu/~rstan/ec/newerr.ps

[39] Stanley, R. P. Enumerative Combinatorics, 2, Cambridge university press, 1999 | MR | Zbl

[40] Verma, D.-N. Review of the paper “Affine root systems and Dedekind’s $\eta$-function" written by Macdonald, I. G., MR0357528(50#9996), MathSciNet, 7 pages

[41] Weisstein, E. W. Elder’s Theorem, from MathWorld – A Wolfram Web Resource

[42] Weisstein, E. W. Stanley’s Theorem, from MathWorld – A Wolfram Web Resource

[43] Winquist, L. An elementary proof of $p(11m+6)\equiv 0\left(\mathrm{mod}11\right)$, J. Combinatorial Theory, Volume 6 (1969), pp. 56-59 | DOI | MR | Zbl

[44] Zeilberger, D. A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof, Discrete Math., Volume 51 (1984), pp. 101-108 | DOI | MR | Zbl