## The sparsity of character tables of high rank groups of Lie type

HTML articles powered by AMS MathViewer

- by Michael J. Larsen and Alexander R. Miller
- Represent. Theory
**25**(2021), 173-192 - DOI: https://doi.org/10.1090/ert/560
- Published electronically: March 4, 2021
- PDF | Request permission

## Abstract:

In the high rank limit, the fraction of non-zero character table entries of finite simple groups of Lie type goes to zero.## References

- R. W. Carter,
*Centralizers of semisimple elements in the finite classical groups*, Proc. London Math. Soc. (3)**42**(1981), no. 1, 1–41. MR**602121**, DOI 10.1112/plms/s3-42.1.1 - Paul Erdös and Joseph Lehner,
*The distribution of the number of summands in the partitions of a positive integer*, Duke Math. J.**8**(1941), 335–345. MR**4841** - P. Erdős and P. Turán,
*On some problems of a statistical group-theory. I*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**4**(1965), 175–186 (1965). MR**184994**, DOI 10.1007/BF00536750 - Jason Fulman and Robert Guralnick,
*Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements*, Trans. Amer. Math. Soc.**364**(2012), no. 6, 3023–3070. MR**2888238**, DOI 10.1090/S0002-9947-2012-05427-4 - Jason Fulman and Robert Guralnick,
*The number of regular semisimple conjugacy classes in the finite classical groups*, Linear Algebra Appl.**439**(2013), no. 2, 488–503. MR**3089699**, DOI 10.1016/j.laa.2013.03.031 - Patrick X. Gallagher,
*Degrees, class sizes and divisors of character values*, J. Group Theory**15**(2012), no. 4, 455–467. MR**2948774**, DOI 10.1515/jgt-2012-0008 - Patrick X. Gallagher, Michael Larsen, and Alexander R. Miller,
*Many zeros of many characters of $\textrm {{GL}}(n,q)$*, Int. Math. Res. Not. IMRN. rnaa160. Published July 10, 2020. DOI:10.1093/imrn/rnaa160. - Samuel Gonshaw, Martin W. Liebeck, and E. A. O’Brien,
*Unipotent class representatives for finite classical groups*, J. Group Theory**20**(2017), no. 3, 505–525. MR**3641686**, DOI 10.1515/jgth-2016-0047 - Michael Larsen and Aner Shalev,
*Fibers of word maps and some applications*, J. Algebra**354**(2012), 36–48. MR**2879221**, DOI 10.1016/j.jalgebra.2011.10.040 - Michael Larsen and Aner Shalev,
*On the distribution of values of certain word maps*, Trans. Amer. Math. Soc.**368**(2016), no. 3, 1647–1661. MR**3449221**, DOI 10.1090/tran/6389 - Frank Lübeck,
*Numbers of Conjugacy Classes in Finite Groups of Lie Type*, 2004, http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/nrclasses/nrclasses.html - G. Lusztig,
*Irreducible representations of finite classical groups*, Invent. Math.**43**(1977), no. 2, 125–175. MR**463275**, DOI 10.1007/BF01390002 - George Lusztig,
*Characters of reductive groups over a finite field*, Annals of Mathematics Studies, vol. 107, Princeton University Press, Princeton, NJ, 1984. MR**742472**, DOI 10.1515/9781400881772 - G. Lusztig,
*On the representations of reductive groups with disconnected centre*, Astérisque**168**(1988), 10, 157–166. Orbites unipotentes et représentations, I. MR**1021495** - 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 - Robert Steinberg,
*Endomorphisms of linear algebraic groups*, Memoirs of the American Mathematical Society, No. 80, American Mathematical Society, Providence, R.I., 1968. MR**0230728**

## Bibliographic Information

**Michael J. Larsen**- Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana
- MR Author ID: 293592
- Email: mjlarsen@indiana.edu
**Alexander R. Miller**- Affiliation: Faculty of Mathematics, University of Vienna, Austria
- MR Author ID: 881590
- Email: alexander.r.miller@univie.ac.at
- Received by editor(s): June 9, 2020
- Received by editor(s) in revised form: November 19, 2020, and December 2, 2020
- Published electronically: March 4, 2021
- Additional Notes: The first author was partially supported by the NSF grant DMS-1702152. The second author was partially supported by the Austrian Science Foundation.
- © Copyright 2021 American Mathematical Society
- Journal: Represent. Theory
**25**(2021), 173-192 - MSC (2020): Primary 20C33
- DOI: https://doi.org/10.1090/ert/560
- MathSciNet review: 4224713