Home Almost all entries in the character table of the symmetric group are multiples of any given prime
Article
Licensed
Unlicensed Requires Authentication

Almost all entries in the character table of the symmetric group are multiples of any given prime

  • Sarah Peluse EMAIL logo and Kannan Soundararajan
Published/Copyright: March 25, 2022

Abstract

We show that almost every entry in the character table of S N is divisible by any fixed prime as N . This proves a conjecture of Miller.

Award Identifier / Grant number: DMS-1903038

Funding statement: The first author is partially supported by the NSF Mathematical Sciences Postdoctoral Research Fellowship Program under Grant No. DMS-1903038 and by the Oswald Veblen Fund. The second author is partially supported by a grant from the National Science Foundation, and a Simons Investigator Grant from the Simons Foundation.

Acknowledgements

We thank the referees for their careful reading.

References

[1] N. G. de Bruijn, On Mahler’s partition problem, Nederl. Akad. Wetensch., Proc. 51 (1948), 659–669 = Indagationes Math. 10 (1948), 210–220. Search in Google Scholar

[2] P. Erdös and J. Lehner, The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8 (1941), 335–345. 10.1215/S0012-7094-41-00826-8Search in Google Scholar

[3] P. Flajolet and R. Sedgewick, Analytic combinatorics, Cambridge University Press, Cambridge 2009. 10.1017/CBO9780511801655Search in Google Scholar

[4] W. Fulton and J. Harris, Representation theory. A first course, Readings in mathematics, Grad. Texts in Math. 129, Springer, New York 1991. Search in Google Scholar

[5] D. Gluck, Parity in columns of the character table of S n , Proc. Amer. Math. Soc. 147 (2019), no. 3, 1005–1011. 10.1090/proc/14300Search in Google Scholar

[6] M. J. Larsen and A. R. Miller, The sparsity of character tables of high rank groups of Lie type, Represent. Theory 25 (2021), 173–192. 10.1090/ert/560Search in Google Scholar

[7] K. Mahler, On a special functional equation, J. Lond. Math. Soc. 15 (1940), 115–123. 10.1112/jlms/s1-15.2.115Search in Google Scholar

[8] A. Malik, F. Stan and A. Zaharescu, The Siegel norm, the length function and character values of finite groups, Indag. Math. (N.S.) 25 (2014), no. 3, 475–486. 10.1016/j.indag.2013.12.001Search in Google Scholar

[9] J. McKay, Irreducible representations of odd degree, J. Algebra 20 (1972), 416–418. 10.1016/0021-8693(72)90066-XSearch in Google Scholar

[10] A. R. Miller, The probability that a character value is zero for the symmetric group, Math. Z. 277 (2014), no. 3–4, 1011–1015. 10.1007/s00209-014-1290-xSearch in Google Scholar

[11] A. R. Miller, On parity and characters of symmetric groups, J. Combin. Theory Ser. A 162 (2019), 231–240. 10.1016/j.jcta.2018.11.001Search in Google Scholar

[12] L. Morotti, On divisibility by primes in columns of character tables of symmetric groups, Arch. Math. (Basel) 114 (2020), no. 4, 361–365. 10.1007/s00013-019-01407-5Search in Google Scholar

[13] A. M. Odlyzko and E. M. Rains, On longest increasing subsequences in random permutations, Analysis, geometry, number theory: The mathematics of Leon Ehrenpreis (Philadelphia 1998), Contemp. Math. 251, American Mathematical Society, Providence (2000), 439–451. 10.1090/conm/251/03886Search in Google Scholar

[14] S. Peluse, On even entries in the character table of the symmetric group, preprint (2020), https://arxiv.org/abs/2007.06652. Search in Google Scholar

[15] H. Rademacher, On the partition function p ( n ) , Proc. Lond. Math. Soc. (2) 43 (1937), no. 4, 241–254. 10.1112/plms/s2-43.4.241Search in Google Scholar

[16] A. M. Vershik and S. V. Kerov, Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25–36, 96. 10.1007/BF01086021Search in Google Scholar

Received: 2020-07-27
Revised: 2021-11-23
Published Online: 2022-03-25
Published in Print: 2022-05-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 28.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/crelle-2022-0004/html
Scroll to top button