On the existence of tableaux with given modular major index
Algebraic Combinatorics, Volume 1 (2018) no. 1, p. 3-21
We provide simple necessary and sufficient conditions for the existence of a standard Young tableau of a given shape and major index r mod n, for all r. Our result generalizes the r=1 case due essentially to Klyachko [11] and proves a recent conjecture due to Sundaram [32] for the r=0 case. A byproduct of the proof is an asymptotic equidistribution result for “almost all” shapes. The proof uses a representation-theoretic formula involving Ramanujan sums and normalized symmetric group character estimates. Further estimates involving “opposite” hook lengths are given which are well-adapted to classifying which partitions λn have f λ n d for fixed d. We also give a new proof of a generalization of the hook length formula due to Fomin-Lulov [4] for symmetric group characters at rectangles. We conclude with some remarks on unimodality of symmetric group characters.
Received : 2017-08-19
Accepted : 2017-08-19
Published online : 2018-01-29
DOI : https://doi.org/10.5802/alco.4
Classification:  05E10
Keywords: Standard Young tableaux, symmetric group characters, major index, hook length formula, rectangular partitions
@article{ALCO_2018__1_1_3_0,
     author = {Swanson, Joshua P.},
     title = {On the existence of tableaux with given modular major index},
     journal = {Algebraic Combinatorics},
     publisher = {MathOA foundation},
     volume = {1},
     number = {1},
     year = {2018},
     pages = {3-21},
     doi = {10.5802/alco.4},
     zbl = {06882332},
     language = {en},
     url = {http://alco.centre-mersenne.org/item/ALCO_2018__1_1_3_0}
}
Swanson, Joshua P. On the existence of tableaux with given modular major index. Algebraic Combinatorics, Volume 1 (2018) no. 1, pp. 3-21. doi : 10.5802/alco.4. https://alco.centre-mersenne.org/item/ALCO_2018__1_1_3_0/

[1] Adin, Ron M.; Brenti, Francesco; Roichman, Yuval Descent representations and multivariate statistics, Trans. Amer. Math. Soc., Volume 357 (2005) no. 8, pp. 3051-3082 | Article | MR 2135735 | Zbl 1059.05105

[2] Ciocan-Fontanine, Ionuţ; Konvalinka, Matjaž; Pak, Igor The weighted hook length formula, J. Combin. Theory Ser. A, Volume 118 (2011) no. 6, pp. 1703-1717 | Article | MR 2793605 | Zbl 1227.05034

[3] Désarmémien, Jacques Étude modulo n des statistiques mahoniennes, Séminaire Lotharingien de Combinatoire, Volume 22 (1990), pp. 27-35 | Zbl 0981.05507

[4] Fomin, Sergey; Lulov, Nathan On the number of rim hook tableaux, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), Volume 223 (1995), pp. 219-226 | Article | MR 1374321 | Zbl 0884.05095

[5] Foulkes, H. O. Characters of symmetric groups induced by characters of cyclic subgroups, Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), Inst. Math. Appl., Southend-on-Sea (1972), pp. 141-154 | MR 342600

[6] Fulton, W. Young Tableaux; with applications to representation theory and geometry, Cambridge University Press, New York, London Mathematical Society Student Texts, Volume 35 (1997) | MR 1464693 | Zbl 0878.14034

[7] Garsia, A. M.; Procesi, C. On certain graded S n -modules and the q-Kostka polynomials, Adv. Math., Volume 94 (1992) no. 1, pp. 82-138 | Article | MR 1168926 | Zbl 0797.20012

[8] James, Gordon; Kerber, Adalbert The representation theory of the symmetric group, Addison-Wesley Publishing Co., Reading, Mass., Encyclopedia of Mathematics and its Applications, Volume 16 (1981), xxviii+510 pages | MR 644144 | Zbl 0491.20010

[9] Johnson, Marianne Standard tableaux and Klyachko’s theorem on Lie representations, J. Combin. Theory Ser. A, Volume 114 (2007) no. 1, pp. 151-158 | Article | MR 2276964 | Zbl 1227.05266

[10] Kerov, S. A q-analog of the hook walk algorithm for random Young tableaux, J. Algebraic Combin., Volume 2 (1993) no. 4, pp. 383-396 | Article | MR 1241507 | Zbl 0785.05087

[11] Klyachko, A. A. Lie elements in the tensor algebra, Siberian Mathematical Journal, Volume 15 (1974) no. 6, pp. 914-920 | Article | MR 371961 | Zbl 0325.15018

[12] Knopfmacher, John Abstract analytic number theory, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, North-Holland Mathematical Library (1975) no. 12, ix+322 pages | MR 419383 | Zbl 0322.10001

[13] Kovács, L. G.; Stöhr, Ralph A combinatorial proof of Klyachko’s theorem on Lie representations, J. Algebraic Combin., Volume 23 (2006) no. 3, pp. 225-230 | Article | MR 2228926 | Zbl 1163.20312

[14] Kraśkiewicz, Witold; Weyman, Jerzy Algebra of coinvariants and the action of a Coxeter element, Bayreuth. Math. Schr. (2001) no. 63, pp. 265-284 | MR 1867283 | Zbl 1037.20012

[15] Lam, T. Y. Young diagrams, Schur functions, the Gale–Ryser theorem and a conjecture of Snapper, J. Pure Appl. Algebra, Volume 10 (1977/78) no. 1, pp. 81-94 | Article | MR 472992 | Zbl 0373.05009

[16] Larsen, Michael; Shalev, Aner Characters of symmetric groups: sharp bounds and applications, Invent. Math., Volume 174 (2008) no. 3, pp. 645-687 | Article | MR 2453603 | Zbl 1166.20009

[17] Liebler, R. A.; Vitale, M. R. Ordering the partition characters of the symmetric group, J. Algebra, Volume 25 (1973), pp. 487-489 | Article | MR 316544 | Zbl 0274.20016

[18] Morales, Alejandro; Pak, Igor; Panova, Greta Asymptotics of the number of standard Young tableaux of skew shape (2017) (Preprint)

[19] Pak, Igor Inequality for hook numbers in Young diagrams, MathOverflow (https://mathoverflow.net/q/243846 (version: 2017-04-13))

[20] Reutenauer, Christophe Free Lie algebras, The Clarendon Press, Oxford University Press, New York, London Mathematical Society Monographs. New Series, Volume 7 (1993), xviii+269 pages (Oxford Science Publications) | MR 1231799 | Zbl 0798.17001

[21] Roichman, Yuval Upper bound on the characters of the symmetric groups, Invent. Math., Volume 125 (1996) no. 3, pp. 451-485 | Article | MR 1400314 | Zbl 0854.20015

[22] Sagan, Bruce E. The symmetric group. Representations, combinatorial algorithms, and symmetric functions, Springer-Verlag, New York, Graduate Texts in Mathematics, Volume 203 (2001), xvi+238 pages | Article | MR 1824028 | Zbl 0964.05070

[23] Schocker, Manfred Embeddings of higher Lie modules, J. Pure Appl. Algebra, Volume 185 (2003) no. 1-3, pp. 279-288 | Article | MR 2006431 | Zbl 1048.20030

[24] Serre, Jean-Pierre Linear representations of finite groups, Springer-Verlag, New York-Heidelberg, Graduate Texts in Mathematics (1977) no. 42, x+170 pages (Translated from the second French edition by Leonard L. Scott) | MR 450380 | Zbl 0355.20006

[25] Snapper, Ernst Group characters and nonnegative integral matrices, J. Algebra, Volume 19 (1971), pp. 520-535 | Article | MR 284523 | Zbl 0226.20008

[26] Spencer, Joel Asymptopia, American Mathematical Society, Providence, RI, Student Mathematical Library, Volume 71 (2014), xiv+183 pages (With Laura Florescu) | Article | MR 3185739 | Zbl 06319745

[27] Springer, T. A. Regular elements of finite reflection groups, Invent. Math., Volume 25 (1974), pp. 159-198 | Article | MR 354894 | Zbl 0287.20043

[28] Stanley, R. P. Enumerative combinatorics. Vol. 2, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Volume 62 (1999), xii+581 pages (With a foreword by Gian Carlo Rota and appendix 1 by Sergey Fomin) | Article | MR 1676282 | Zbl 0978.05002

[29] Stanley, Richard P. Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. (N.S.), Volume 1 (1979) no. 3, pp. 475-511 | Article | MR 526968 | Zbl 0497.20002

[30] Stanley, Richard P. The stable behavior of some characters of SL (n,C), Linear and Multilinear Algebra, Volume 16 (1984) no. 1-4, pp. 3-27 | Article | MR 768993 | Zbl 0573.20042

[31] Stembridge, John R. On the eigenvalues of representations of reflection groups and wreath products, Pacific J. Math., Volume 140 (1989) no. 2, pp. 353-396 | Article | MR 1023791 | Zbl 0641.20011

[32] Sundaram, Sheila On conjugacy classes of S n containing all irreducibles (2017) (to appear in Israel J. Math.)