Table des matières de ce fascicule | Article suivant
Guo-Niu Han
The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
(La formule de longueur d’équerre de Nekrasov-Okounkov : raffinement, démonstration élémentaire, extension et applications)
Annales de l'institut Fourier, 60 no. 1 (2010), p. 1-29
Article: sur abonnement
Class. Math.: 05A15, 05A17, 05A19, 11P82, 17B22
Mots clés: longueur d’équerre, formule d’équerre, partition, $t$-core, produit d’Euler, identités de Macdonald
[1] R. M. Adin & A. Frumkin, “Rim Hook Tableaux and Kostant’s $\eta $-Function Coefficients”, Adv. in Appl. Math. 33 (2004), p. 492-511 MR 2081040 | Zbl 1056.05144
[2] G. E. Andrews, The Theory of Partitions, Encyclopedia of Math. and Its Appl. 2, Addison-Wesley, Reading, 1976 MR 557013 | Zbl 0371.10001
[3] R. Bacher & L. Manivel, “Hooks and Powers of Parts in Partitions”, article B47d, 11 pages, 2001
arXiv | MR 1894024 | Zbl 1021.05008
[4] A. Berkovich & F. G. Garvan, “The BG-rank of a partition and its applications”, Adv. in Appl. Math. 40 (2008), p. 377-400 MR 2402176 | Zbl pre05268092
[5] C. Bessenrodt, “On hooks of Young diagrams”, Ann. of Comb. 2 (1998), p. 103-110 MR 1682922 | Zbl 0929.05091
[6] E. Carlsson & A. Okounkov, “Exts and Vertex Operators”
arXiv
[7] P. Cellini, P. M. Frajria & P. Papi, “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 208 (2007), p. 1103-1119 MR 2283450 | Zbl 1160.17007
[8] F. J. Dyson, “Missed opportunities”, Bull. Amer. Math. Soc. 78 (1972), p. 635-652
Article | MR 522147 | Zbl 0271.01005
[9] L. Euler, “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”, on arXiv:math.HO/0411454
[10] H. M. Farkas & I. Kra, “On the Quintuple Product Identity”, Proc. Amer. Math. Soc. 27 (1999), p. 771-778 MR 1487364 | Zbl 0932.11029
[11] D. Foata & G.-N. Han, “The triple, quintuple and septuple product identities revisited”, Art. B42o, 12 pp Zbl 0923.11143
[12] J. S. Frame, G. de Beauregard Robinson & R. M. Thrall, “The hook graphs of the symmetric groups”, Canadian J. Math. 6 (1954), p. 316-324 MR 62127 | Zbl 0055.25404
[13] F. Garvan, D. Kim & D. Stanton, “Cranks and $t$-cores”, Invent. Math. 101 (1990), p. 1-17 MR 1055707 | Zbl 0721.11039
[14] I. Gessel & G. Viennot, “Binomial determinants, paths, and hook length formulae”, Adv. in Math. 58 (1985), p. 300-321 MR 815360 | Zbl 0579.05004
[15] C. Greene, A. Nijenhuis & H. S. Wilf, “A probabilistic proof of a formula for the number of Young tableaux of a given shape”, Adv. in Math. 31 (1979), p. 104-109 MR 521470 | Zbl 0398.05008
[16] G.-N. Han, “An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths”, 35 pages, 2008
arXiv
[17] G.-N. Han, “Discovering hook length formulas by an expansion technique”, Electron. J. Combin., vol.
[18] A. Hoare & M. Howard, “An Involution of Blocks in the Partitions of $n$”, Amer. Math. Monthly 93 (1986), p. 475-476 MR 843195 | Zbl 0611.10006
[19] G. James & A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing, Reading, MA, 1981 MR 644144 | Zbl 0491.20010
[20] J. T. Joichi & D. Stanton, “An involution for Jacobi’s identity”, Discrete Math. 73 (1989), p. 261-271 MR 983024 | Zbl 0661.05007
[21] V. G. Kac, “Infinite-dimensional Lie algebras and Dedekind’s $\eta $-function”, Functional Anal. Appl. 8 (1974), p. 68-70 MR 374210 | Zbl 0299.17005
[22] M. S. Kirdar & T. H. R. Skyrme, “On an Identity Related to Partitions and Repetitions of Parts”, Canad. J. Math. 34 (1982), p. 194-195 MR 650858 | Zbl 0482.10013
[23] D. E. Knuth, The Art of Computer Programming, Sorting and Searching, 2nd ed. 3, Addison Wesley Longman, 1998 MR 378456
[24] B. Kostant, “On Macdonald’s $\eta $-function formula, the Laplacian and generalized exponents”, Adv. in Math. 20 (1976), p. 179-212 MR 485661 | Zbl 0339.10019
[25] B. Kostant, “Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra”, Invent. Math. 158 (2004), p. 181-226 MR 2090363 | Zbl 1076.17002
[26] C. Krattenthaler, “Another involution principle-free bijective proof of Stanley’s hook-content formula”, J. Combin. Theory Ser. A 88 (1999), p. 66-92 MR 1713492 | Zbl 0936.05087
[27] Alain Lascoux, 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 2017492 | Zbl 1039.05066
[28] I. G. Macdonald, “Affine root systems and Dedekind’s $\eta $-function”, Invent. Math. 15 (1972), p. 91-143 MR 357528 | Zbl 0244.17005
[29] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Clarendon Press, Oxford, 1995 MR 1354144 | Zbl 0824.05059
[30] S. C. Milne, “An elementary proof of the Macdonald identities for $A^{(1)}_l$”, Adv. in Math. 57 (1985), p. 34-70 MR 800859 | Zbl 0586.33011
[31] R. V. Moody, “Macdonald identities and Euclidean Lie algebras”, Proc. Amer. Math. Soc. 48 (1975), p. 43-52 MR 442048 | Zbl 0315.17003
[32] N. A. Nekrasov & A. Okounkov, Seiberg-Witten theory and random partitions. The unity of mathematics, 2006, p. 525-596 MR 2181816 | Zbl pre05050087
[33] J.-C. Novelli, I. Pak & A. V. Stoyanovskii, “A direct bijective proof of the hook-length formula”, Discrete Math. Theor. Comput. Sci. 1 (1997), p. 53-67 MR 1605030 | Zbl 0934.05125
[34] J. B. Remmel & R. Whitney, “A bijective proof of the hook formula for the number of column strict tableaux with bounded entries”, European J. Combin. 4 (1983), p. 45-63 MR 694468 | Zbl 0521.05007
[35] H. Rosengren & M. Schlosser, “Elliptic determinant evaluations and the Macdonald identities for affine root systems”, Compositio Math. 142 (2006), p. 937-961 MR 2249536 | Zbl 1104.15009
[36] J.-P. Serre, Cours d’arithmétique, Collection SUP: “Le Mathématicien”, 2 Presses Universitaires de France, Paris, 1970 MR 255476 | Zbl 0225.12002
[37] N. Sloane & al., “The On-Line Encyclopedia of Integer Sequences”, http:// www.research.att.com/~njas/sequences/
arXiv | Zbl 1044.11108
[38] R. P. Stanley, “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] R. P. Stanley, Enumerative Combinatorics 2, Cambridge university press, 1999 MR 1676282 | Zbl 0928.05001
[40] D.-N. Verma, “Review of the paper “Affine root systems and Dedekind’s $\eta $-function" written by Macdonald”
[41] E. W. Weisstein, “Elder’s Theorem, from MathWorld – A Wolfram Web Resource”
[42] E. W. Weisstein, “Stanley’s Theorem, from MathWorld – A Wolfram Web Resource”
[43] L. Winquist, “An elementary proof of $p(11m+6)\equiv 0\,({\rm mod} 11)$”, J. Combinatorial Theory 6 (1969), p. 56-59 MR 236136 | Zbl 0241.05006
[44] D. Zeilberger, “A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof”, Discrete Math. 51 (1984), p. 101-108 MR 755045 | Zbl 0551.05010
Guo-Niu Han
The Nekrasov-Okounkov hook length formula: refinement, elementary proof, extension and applications
(La formule de longueur d’équerre de Nekrasov-Okounkov : raffinement, démonstration élémentaire, extension et applications)
Annales de l'institut Fourier, 60 no. 1 (2010), p. 1-29
Article: sur abonnement
Class. Math.: 05A15, 05A17, 05A19, 11P82, 17B22
Mots clés: longueur d’équerre, formule d’équerre, partition, $t$-core, produit d’Euler, identités de Macdonald
Résumé - Abstract
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”.
Bibliographie
[2] G. E. Andrews, The Theory of Partitions, Encyclopedia of Math. and Its Appl. 2, Addison-Wesley, Reading, 1976 MR 557013 | Zbl 0371.10001
[3] R. Bacher & L. Manivel, “Hooks and Powers of Parts in Partitions”, article B47d, 11 pages, 2001
arXiv | MR 1894024 | Zbl 1021.05008
[4] A. Berkovich & F. G. Garvan, “The BG-rank of a partition and its applications”, Adv. in Appl. Math. 40 (2008), p. 377-400 MR 2402176 | Zbl pre05268092
[5] C. Bessenrodt, “On hooks of Young diagrams”, Ann. of Comb. 2 (1998), p. 103-110 MR 1682922 | Zbl 0929.05091
[6] E. Carlsson & A. Okounkov, “Exts and Vertex Operators”
arXiv
[7] P. Cellini, P. M. Frajria & P. Papi, “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 208 (2007), p. 1103-1119 MR 2283450 | Zbl 1160.17007
[8] F. J. Dyson, “Missed opportunities”, Bull. Amer. Math. Soc. 78 (1972), p. 635-652
Article | MR 522147 | Zbl 0271.01005
[9] L. Euler, “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”, on arXiv:math.HO/0411454
[10] H. M. Farkas & I. Kra, “On the Quintuple Product Identity”, Proc. Amer. Math. Soc. 27 (1999), p. 771-778 MR 1487364 | Zbl 0932.11029
[11] D. Foata & G.-N. Han, “The triple, quintuple and septuple product identities revisited”, Art. B42o, 12 pp Zbl 0923.11143
[12] J. S. Frame, G. de Beauregard Robinson & R. M. Thrall, “The hook graphs of the symmetric groups”, Canadian J. Math. 6 (1954), p. 316-324 MR 62127 | Zbl 0055.25404
[13] F. Garvan, D. Kim & D. Stanton, “Cranks and $t$-cores”, Invent. Math. 101 (1990), p. 1-17 MR 1055707 | Zbl 0721.11039
[14] I. Gessel & G. Viennot, “Binomial determinants, paths, and hook length formulae”, Adv. in Math. 58 (1985), p. 300-321 MR 815360 | Zbl 0579.05004
[15] C. Greene, A. Nijenhuis & H. S. Wilf, “A probabilistic proof of a formula for the number of Young tableaux of a given shape”, Adv. in Math. 31 (1979), p. 104-109 MR 521470 | Zbl 0398.05008
[16] G.-N. Han, “An explicit expansion formula for the powers of the Euler Product in terms of partition hook lengths”, 35 pages, 2008
arXiv
[17] G.-N. Han, “Discovering hook length formulas by an expansion technique”, Electron. J. Combin., vol.
15(1), Research Paper #R133, 41 pp, 2008 MR 2448883 | Zbl 1165.05305[18] A. Hoare & M. Howard, “An Involution of Blocks in the Partitions of $n$”, Amer. Math. Monthly 93 (1986), p. 475-476 MR 843195 | Zbl 0611.10006
[19] G. James & A. Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications 16, Addison-Wesley Publishing, Reading, MA, 1981 MR 644144 | Zbl 0491.20010
[20] J. T. Joichi & D. Stanton, “An involution for Jacobi’s identity”, Discrete Math. 73 (1989), p. 261-271 MR 983024 | Zbl 0661.05007
[21] V. G. Kac, “Infinite-dimensional Lie algebras and Dedekind’s $\eta $-function”, Functional Anal. Appl. 8 (1974), p. 68-70 MR 374210 | Zbl 0299.17005
[22] M. S. Kirdar & T. H. R. Skyrme, “On an Identity Related to Partitions and Repetitions of Parts”, Canad. J. Math. 34 (1982), p. 194-195 MR 650858 | Zbl 0482.10013
[23] D. E. Knuth, The Art of Computer Programming, Sorting and Searching, 2nd ed. 3, Addison Wesley Longman, 1998 MR 378456
[24] B. Kostant, “On Macdonald’s $\eta $-function formula, the Laplacian and generalized exponents”, Adv. in Math. 20 (1976), p. 179-212 MR 485661 | Zbl 0339.10019
[25] B. Kostant, “Powers of the Euler product and commutative subalgebras of a complex simple Lie algebra”, Invent. Math. 158 (2004), p. 181-226 MR 2090363 | Zbl 1076.17002
[26] C. Krattenthaler, “Another involution principle-free bijective proof of Stanley’s hook-content formula”, J. Combin. Theory Ser. A 88 (1999), p. 66-92 MR 1713492 | Zbl 0936.05087
[27] Alain Lascoux, 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 2017492 | Zbl 1039.05066
[28] I. G. Macdonald, “Affine root systems and Dedekind’s $\eta $-function”, Invent. Math. 15 (1972), p. 91-143 MR 357528 | Zbl 0244.17005
[29] I. G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Clarendon Press, Oxford, 1995 MR 1354144 | Zbl 0824.05059
[30] S. C. Milne, “An elementary proof of the Macdonald identities for $A^{(1)}_l$”, Adv. in Math. 57 (1985), p. 34-70 MR 800859 | Zbl 0586.33011
[31] R. V. Moody, “Macdonald identities and Euclidean Lie algebras”, Proc. Amer. Math. Soc. 48 (1975), p. 43-52 MR 442048 | Zbl 0315.17003
[32] N. A. Nekrasov & A. Okounkov, Seiberg-Witten theory and random partitions. The unity of mathematics, 2006, p. 525-596 MR 2181816 | Zbl pre05050087
[33] J.-C. Novelli, I. Pak & A. V. Stoyanovskii, “A direct bijective proof of the hook-length formula”, Discrete Math. Theor. Comput. Sci. 1 (1997), p. 53-67 MR 1605030 | Zbl 0934.05125
[34] J. B. Remmel & R. Whitney, “A bijective proof of the hook formula for the number of column strict tableaux with bounded entries”, European J. Combin. 4 (1983), p. 45-63 MR 694468 | Zbl 0521.05007
[35] H. Rosengren & M. Schlosser, “Elliptic determinant evaluations and the Macdonald identities for affine root systems”, Compositio Math. 142 (2006), p. 937-961 MR 2249536 | Zbl 1104.15009
[36] J.-P. Serre, Cours d’arithmétique, Collection SUP: “Le Mathématicien”, 2 Presses Universitaires de France, Paris, 1970 MR 255476 | Zbl 0225.12002
[37] N. Sloane & al., “The On-Line Encyclopedia of Integer Sequences”, http:// www.research.att.com/~njas/sequences/
arXiv | Zbl 1044.11108
[38] R. P. Stanley, “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] R. P. Stanley, Enumerative Combinatorics 2, Cambridge university press, 1999 MR 1676282 | Zbl 0928.05001
[40] D.-N. Verma, “Review of the paper “Affine root systems and Dedekind’s $\eta $-function" written by Macdonald”
[41] E. W. Weisstein, “Elder’s Theorem, from MathWorld – A Wolfram Web Resource”
[42] E. W. Weisstein, “Stanley’s Theorem, from MathWorld – A Wolfram Web Resource”
[43] L. Winquist, “An elementary proof of $p(11m+6)\equiv 0\,({\rm mod} 11)$”, J. Combinatorial Theory 6 (1969), p. 56-59 MR 236136 | Zbl 0241.05006
[44] D. Zeilberger, “A short hook-lengths bijection inspired by the Greene-Nijenhuis-Wilf proof”, Discrete Math. 51 (1984), p. 101-108 MR 755045 | Zbl 0551.05010
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