Large-scale Monte Carlo simulations for zeros in character tables of symmetric groups

Miller, Alexander; Scheinerman, Danny

doi:10.1090/mcom/3964

Large-scale Monte Carlo simulations for zeros in character tables of symmetric groups
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by Alexander Rossi Miller and Danny Scheinerman;

Math. Comp. 94 (2025), 505-515

DOI: https://doi.org/10.1090/mcom/3964

Published electronically: August 29, 2024

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Abstract:

This is a brief report on some recent large-scale Monte Carlo simulations for approximating the density of zeros in character tables of large symmetric groups. Previous computations suggested that a large fraction of zeros cannot be explained by classical vanishing results. Our computations eclipse previous ones and suggest that the opposite is true. In fact, we find empirically that almost all of the zeros are of a single classical type.

References

Robert L. Bivins, N. Metropolis, Paul R. Stein, and Mark B. Wells, Characters of the symmetric groups of degree $15$ and $16$, Math. Tables Aids Comput. 8 (1954), 212–216. MR 64776, DOI 10.1090/S0025-5718-1954-0064776-8
Stig Comét, Improved methods to calculate the characters of the symmetric group, Math. Comp. 14 (1960), 104–117. MR 119451, DOI 10.1090/S0025-5718-1960-0119451-0
G. Frobenius, Über die Charaktere der symmetrischen Gruppe, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1900, pp. 516–534.
Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, MA, 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
Kôiti Kondô, Table of characters of the symmetric group of degree 14, Proc. Phys.-Math. Soc. Japan (3) 22 (1940), 585–593. MR 2129
D. E. Littlewood, Group Characters and the Structure of Groups, Proc. London Math. Soc. (2) 39 (1935), no. 2, 150–199. MR 1576896, DOI 10.1112/plms/s2-39.1.150
D. E. Littlewood and A. R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. A 233 (1934), 99–141.
Eleanor McSpirit and Ken Ono, Zeros in the character tables of symmetric groups with an $\ell$-core index, Canad. Math. Bull. 66 (2023), no. 2, 467–476. MR 4584475, DOI 10.4153/S0008439522000443
Alexander R. Miller, The probability that a character value is zero for the symmetric group, Math. Z. 277 (2014), no. 3-4, 1011–1015. MR 3229977, DOI 10.1007/s00209-014-1290-x
Alexander R. Miller, On parity and characters of symmetric groups, J. Combin. Theory Ser. A 162 (2019), 231–240. MR 3874600, DOI 10.1016/j.jcta.2018.11.001
F. D. Murnaghan, The Characters of the Symmetric Group, Amer. J. Math. 59 (1937), no. 4, 739–753. MR 1507276, DOI 10.2307/2371341
Tadasi Nakayama, On some modular properties of irreducible representations of a symmetric group, I, Jpn. J. Math. 17 (1940), 165–184., DOI 10.4099/jjm1924.17.0_165
Sarah Peluse and Kannan Soundararajan, Almost all entries in the character table of the symmetric group are multiples of any given prime, J. Reine Angew. Math. 786 (2022), 45–53. MR 4434750, DOI 10.1515/crelle-2022-0004
S. Peluse and K. Soundararajan, Divisibility of character values of the symmetric group by prime powers, arXiv:2301.02203, 2023.
M. Zia-ud-din, The Characters of the Symmetric Group of Order 11!, Proc. London Math. Soc. (2) 39 (1935), no. 3, 200–204. MR 1576897, DOI 10.1112/plms/s2-39.1.200
M. Zia-ud-din, The Characters of the Symmetric Group of Degrees 12 and 13, Proc. London Math. Soc. (2) 42 (1937), no. 5, 340–355. MR 1577036, DOI 10.1112/plms/s2-42.1.340