×

Bounds on the largest Kronecker and induced multiplicities of finite groups. (English) Zbl 1477.20016

Summary: We give new bounds and asymptotic estimates on the largest Kronecker and induced multiplicities of finite groups. The results apply to large simple groups of Lie type and other groups with few conjugacy classes.

MSC:

20C15 Ordinary representations and characters
20C30 Representations of finite symmetric groups
20C33 Representations of finite groups of Lie type
05E10 Combinatorial aspects of representation theory
05E16 Combinatorial aspects of groups and algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Baik, J.; Deift, P.; Johansson, K., On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc, 12, 1119-1178 (1999) · Zbl 0932.05001
[2] Baumeister, B.; Maróti, A.; Tong-Viet, H. P., Finite groups have more conjugacy classes, Forum Math, 29, 259-275 (2017) · Zbl 1377.20023
[3] Biane, P., Representations of symmetric groups and free probability, Adv. Math, 138, 1, 126-181 (1998) · Zbl 0927.20008 · doi:10.1006/aima.1998.1745
[4] Bivins, R. L.; Metropolis, N.; Stein, P. R.; Wells, M. B., Characters of the symmetric groups of degree 15 and 16, Math. Tables Other Aids Comput, 8, 48, 212 (1954) · Zbl 0056.25801 · doi:10.2307/2002094
[5] Bufetov, A. I., On the vershik-Kerov conjecture concerning the Shannon-McMillan-Breiman theorem for the plancherel family of measures on the space of young diagrams, Geom. Funct. Anal, 22, 4, 938-975 (2012) · Zbl 1254.05024 · doi:10.1007/s00039-012-0169-4
[6] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Eynsham, UK: Oxford University Press, Eynsham, UK · Zbl 0568.20001
[7] Craven, D. A., Symmetric group character degrees and hook numbers, Proc. Lond. Math. Soc, 96, 1, 26-50 (2008) · Zbl 1165.20008 · doi:10.1112/plms/pdm028
[8] Diaconis, D.; Isaacs, I. M., Supercharacters and superclasses for algebra groups, Trans. Amer. Math. Soc, 360, 5, 2359-2392 (2007) · Zbl 1137.20008 · doi:10.1090/S0002-9947-07-04365-6
[9] Feit, W.; Fine, N. J., Pairs of commuting matrices over a finite field, Duke Math. J, 27, 1, 91-94 (1960) · Zbl 0097.00702 · doi:10.1215/S0012-7094-60-02709-5
[10] Fulman, J.; Guralnick, R.; 10, Bounds on the number and sizes of conjugacy classes in finite chevalley groups with applications to derangements, Trans. Amer. Math. Soc, 364, 6, 3023-3070 (2012) · Zbl 1256.20048 · doi:10.1090/S0002-9947-2012-05427-4
[11] Fulton, W., Eigenvalues, invariant factors, highest weights, and schubert calculus, Bull. Amer. Math. Soc, 37, 3, 209-249 (2000) · Zbl 0994.15021 · doi:10.1090/S0273-0979-00-00865-X
[12] Gallagher, P., The number of conjugacy classes of a finite group, Math. Z, 118, 3, 175-179 (1970) · Zbl 0221.20006 · doi:10.1007/BF01113339
[13] Garonzi, M.; Maróti, A., On the number of conjugacy classes of a permutation group, J. Combin. Theory, Ser. A, 133, 251-260 (2015) · Zbl 1328.20003 · doi:10.1016/j.jcta.2015.02.007
[14] Guralnick, R.; Robinson, G., On the commuting probability in finite groups, J. Algebra, 300, 2, 509-528 (2006) · Zbl 1100.20045 · doi:10.1016/j.jalgebra.2005.09.044
[15] Halasi, Z.; Hannusch, C.; Nguyen, H. N., The largest character degrees of the symmetric and alternating groups, Proc. Amer. Math. Soc, 144, 5, 1947-1960 (2015) · Zbl 1344.20020 · doi:10.1090/proc/12920
[16] Heide, G.; Saxl, J.; Tiep, P. H.; Zalesski, A. E., Conjugacy action, induced representations and the Steinberg square for simple groups of lie type, Proc. LMS, 106, 908-930 (2013) · Zbl 1372.20017 · doi:10.1112/plms/pds062
[17] Higman, G., Enumerating p-groups. I: inequalities, Proc. LMS, 10, 24-30 (1960) · Zbl 0093.02603 · doi:10.1112/plms/s3-10.1.24
[18] Hung, N. N.; Lewis, L. M.; Schaeffer Fry, A. A., Finite groups with an irreducible character of large degree, Manuscripta Math, 149, 3-4, 523-546 (2016) · Zbl 1344.20012 · doi:10.1007/s00229-015-0793-z
[19] Ikenmeyer, C., The saxl conjecture and dominance order, Disc. Math, 338, 11, 1970-1975 (2015) · Zbl 1314.05223 · doi:10.1016/j.disc.2015.04.027
[20] Isaacs, I. M., Counting characters of upper triangular groups, J. Algebra, 315, 2, 698-719 (2007) · Zbl 1127.20031 · doi:10.1016/j.jalgebra.2007.01.027
[21] Isaacs, I. M., Bounding the order of a group with a large character degree, J. Algebra, 348, 1, 264-275 (2011) · Zbl 1242.20011 · doi:10.1016/j.jalgebra.2011.08.037
[22] Jaikin-Zapirain, A., On the number of conjugacy classes of finite nilpotent groups, Adv. Math, 227, 3, 1129-1143 (2011) · Zbl 1227.20014 · doi:10.1016/j.aim.2011.02.021
[23] Kazarin, L. S.; Sagirov, I. A., On degrees of irreducible characters of finite simple groups, Proc. Steklov Inst. Math, 2, S71-S81 (2001) · Zbl 1121.20301
[24] Keller, T. M., Finite groups have even more conjugacy classes, Isr. J. Math. Math, 181, 1, 433-444 (2011) · Zbl 1261.20030 · doi:10.1007/s11856-011-0017-5
[25] Kerov, S.; Pass, A. M., Representations of symmetric groups with maximal dimension, J. Math. Sci. Math, 59, 5, 1131-1135 (1992) · doi:10.1007/BF01480698
[26] Kovács, L.; Robinson, G., On the number of conjugacy classes of a finite group, J. Algebra, 160, 441-460 (1993) · Zbl 0830.20048 · doi:10.1006/jabr.1993.1196
[27] Kowalski, E., The Large Sieve and Its Applications. Arithmetic Geometry, random Walks and Discrete Groups, 293 (2008), Cambridge University Press · Zbl 1177.11080
[28] Larsen, M.; Malle, G.; Tiep, P. H., The largest irreducible representations of simple groups, Proc. LMS, 106, 65-96 (2013) · Zbl 1319.20013 · doi:10.1112/plms/pds030
[29] Liebeck, M.; Pyber, L., Upper bounds for the number of conjugacy classes of a finite group, J. Algebra, 198, 2, 538-562 (1997) · Zbl 0892.20017 · doi:10.1006/jabr.1997.7158
[30] Logan, B. F.; Shepp, L. A., A variational problem for random young tableaux, Adv. Math, 26, 2, 206-222 (1977) · Zbl 0363.62068 · doi:10.1016/0001-8708(77)90030-5
[31] Luo, S.; Sellke, M., The saxl conjecture for fourth powers via the semigroup property, J. Algebr. Comb, 45, 1, 33-80 (2017) · Zbl 1355.05269 · doi:10.1007/s10801-016-0700-z
[32] Maslen, D. K.; Rockmore, D. N., Separation of variables and the computation of fourier transforms on finite groups. I, J. AMS, 10, 169-214 (1997) · Zbl 0860.20016
[33] Mckay, J., The largest degrees of irreducible characters of the symmetric group, Math. Comp. Comp, 30, 135, 624-631 (1976) · Zbl 0345.20011 · doi:10.1090/S0025-5718-1976-0404414-X
[34] Moretó, A., Complex group algebras of finite groups: Brauer’s problem 1, El. Res. Announc. AMS, 11, 34-39 (2005) · Zbl 1077.20003
[35] Pak, I., Randomization Methods in Algorithm Design, When and how n choose k, 191-238 (1999), Providence, RI: AMS, Providence, RI
[36] Pak, I.; Panova, G.; Vallejo, E., Kronecker products, characters, partitions, and the tensor square conjectures, Adv. Math, 288, 702-731 (2016) · Zbl 1328.05199 · doi:10.1016/j.aim.2015.11.002
[37] Pak, I., Panova, G., Yeliussizov, D. On the largest Kronecker and Littlewood-Richardson coefficients. arXiv:1804.04693. · Zbl 1414.05305
[38] Pyber, L., Finite groups have many conjugacy classes, J. LMS, 46, 239-249 (1992) · Zbl 0712.20016 · doi:10.1112/jlms/s2-46.2.239
[39] Romik, D., The Surprising Mathematics of Longest Increasing Subsequences (2015), New York: Cambridge University Press, New York · Zbl 1345.05003
[40] Sherman, G., A lower bound for the number of conjugacy classes in a finite nilpotent group, Pacific J. Math, 80, 1, 253-254 (1979) · Zbl 0377.20017 · doi:10.2140/pjm.1979.80.253
[41] Snyder, N., Groups with a character of large degree, Proc. Amer. Math. Soc, 136, 6, 1893-1903 (2008) · Zbl 1145.20005 · doi:10.1090/S0002-9939-08-09147-8
[42] Soffer, A., Upper bounds on the number of conjugacy classes in unitriangular groups, J. Group Theory, 19, 1063-1095 (2016) · Zbl 1352.20036
[43] Stanley, R. P., Enumerative Combinatorics, Vol.1 (2012), Cambridge University · Zbl 1247.05003
[44] Van Leeuwen, M. A. A., Interaction of Combinatorics and Representation Theory, The Littlewood-Richardson rule, and related combinatorics, 95-145 (2001), Tokyo: Math. Soc, Tokyo · Zbl 0991.05101
[45] Vershik, A. M.; Kerov, S. V., The asymptotic character theory of the symmetric group, Funct. Anal. Its Appl, 15, 4, 246-255 (1982) · Zbl 0507.20006 · doi:10.1007/BF01106153
[46] Vershik, A. M.; Kerov, S. V., Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group, Funct. Anal. Appl, 19, 1, 21-31 (1985) · Zbl 0592.20015 · doi:10.1007/BF01086021
[47] Vershik, A. M.; Pavlov, D., Numerical experiments in problems of asymptotic representation theory, J. Math. Sci, 168, 3, 351-361 (2010) · Zbl 1288.20012 · doi:10.1007/s10958-010-9986-x
[48] Vinroot, C. R., Twisted Frobenius-Schur indicators of finite symplectic groups, J. Algebra, 293, 1, 279-311 (2005) · Zbl 1093.20026 · doi:10.1016/j.jalgebra.2005.01.023
[49] Yan, N., Representation Theory of the Finite Unipotent Linear Groups, 41 (2001), University of Pennsylvania
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.