Geodesic
The fields of values of characters of degree not divisible by p
Forum of Mathematics, Pi, Tome 9 (2021)

Voir la notice de l'article provenant de la source Cambridge University Press

We study the fields of values of the irreducible characters of a finite group of degree not divisible by a prime p. In the case where $p=2$, we fully characterise these fields. In order to accomplish this, we generalise the main result of [ILNT] to higher irrationalities. We do the same for odd primes, except that in this case the analogous results hold modulo a simple-to-state conjecture on the character values of quasi-simple groups. 
@article{10_1017_fmp_2021_1,
     author = {Gabriel Navarro and Pham Huu Tiep},
     title = {The fields of values of characters of degree not divisible by p},
     journal = {Forum of Mathematics, Pi},
     publisher = {mathdoc},
     volume = {9},
     year = {2021},
     doi = {10.1017/fmp.2021.1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.1/}
}
TY  - JOUR
AU  - Gabriel Navarro
AU  - Pham Huu Tiep
TI  - The fields of values of characters of degree not divisible by p
JO  - Forum of Mathematics, Pi
PY  - 2021
VL  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.1/
DO  - 10.1017/fmp.2021.1
LA  - en
ID  - 10_1017_fmp_2021_1
ER  - 
%0 Journal Article
%A Gabriel Navarro
%A Pham Huu Tiep
%T The fields of values of characters of degree not divisible by p
%J Forum of Mathematics, Pi
%D 2021
%V 9
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1017/fmp.2021.1/
%R 10.1017/fmp.2021.1
%G en
%F 10_1017_fmp_2021_1
Gabriel Navarro; Pham Huu Tiep. The fields of values of characters of degree not divisible by p. Forum of Mathematics, Pi, Tome 9 (2021). doi: 10.1017/fmp.2021.1

[Atlas] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., An ATLAS of Finite Groups (Clarendon Press, Oxford, 1985).Google Scholar

[C] Carter, R., Finite Groups of Lie type: Conjugacy Classes and Complex Characters (Wiley, Chichester, 1985).Google Scholar

[D1] Dade, E. C., ‘Blocks with cyclic defect groups’, Ann. of Math. 84 (1966), 20–48.Google Scholar | DOI

[D3] Dade, E. C., ‘Counting characters in blocks with cyclic defect groups, I’, J. Algebra 186 (1996), 934–969.Google Scholar | DOI

[DM] Digne, F. and Michel, J., Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts, 21 (Cambridge University Press, Cambridge, 1991).Google Scholar

[FG] Fein, B. and Gordon, B., ‘Fields generated by characters of finite groups’, J. Lond. Math. Soc. 4 (1972), 735–740.Google Scholar | DOI

[GAP] The GAP Group, ‘GAP – groups, algorithms, and programming’, version 4.10.0 (2018). URL: http://www.gap-system.org.Google Scholar

[Gi] Giannelli, E., ‘The Sylow restriction refinement of the McKay conjecture for symmetric and alternating groups’, Algebra Number Theory, 2021 (to appear).Google Scholar

[GKNT] Giannelli, E., Kleshchev, A. S., Navarro, G. and Tiep, P. H., ‘Restriction of odd degree characters and natural correspondences’, Int. Math. Res. Not. IMRN 2017 (20), 6089–6118.Google Scholar

[GLBST] Guralnick, R. M., Liebeck, M. W., O’Brien, E., Shalev, A. and Tiep, P. H, ‘Surjective word maps and Burnside’s theorem’, Invent. Math. 213 (2018), 589–695.Google Scholar | DOI

[GM] Geck, M. and Malle, G., The Character Theory of Finite Groups of Lie Type: A Guided Tour, Cambridge Studies in Advanced Mathematics (Cambridge University Press, 2020).Google Scholar | DOI

[GT] Guralnick, R. M. and Tiep, P. H., ‘Low-dimensional representations of special linear groups in cross characteristic’, Proc. Lond. Math. Soc. 78 (1999), 116–138.Google Scholar | DOI

[HH] Hoffman, P. N. and Humphreys, J. F., Projective Representations of the Symmetric Group (Clarendon Press, Oxford, 1992).Google Scholar

[ILNT] Isaacs, I. M., Liebeck, M., Navarro, G. and Tiep, P. H, ‘Fields of values of odd degree irreducible characters’, Adv. Math. 354 (2019), 106757.Google Scholar | DOI

[Is1] Isaacs, I. M, ‘Characters of solvable and symplectic groups’, Amer. J. Math. 85 (1973), 594–635.Google Scholar | DOI

[Is2] Isaacs, I. M., Character Theory of Finite Groups (American Mathematical Society, Providence, RI, 2008).Google Scholar

[Is3] Isaacs, I. M., Characters of Solvable Groups (American Mathematical Society, Providence, RI, 2018).Google Scholar | DOI

[JK] James, G. and Kerber, A., The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and Its Applications, 16 (Addison-Wesley, Reading, MA, 1981).Google Scholar

[Ko] Kondrat’Ev, A. S., ‘Normalizers of the Sylow -subgroups in finite simple groups’, Math. Notes 78 (2005), 338–346.Google Scholar | DOI

[LSS] Liebeck, M. W., Saxl, J. and Seitz, G., ‘Subgroups of maximal rank in finite exceptional groups of Lie type’, Proc. Lond. Math. Soc. 65 (1992), 297–325.Google Scholar | DOI

[Lu] Lübeck, F., ‘Character degrees and their multiplicities for some groups of Lie type of rank ’, URL: http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/index.html.Google Scholar

[M] Malle, G., ‘Height characters of finite groups of Lie type’, Represent. Theory 11 (2007), 192–220.Google Scholar | DOI

[MS] Malle, G. and Späth, B., ‘Characters of odd degree’, Ann. of Math. 184 (2016), 869–908.Google Scholar | DOI

[N1] Navarro, G., ‘The McKay conjecture and Galois automorphisms’, Ann. of Math. 160 (2004), 1129–1140.Google Scholar | DOI

[N2] Navarro, G., Character Theory and the McKay Conjecture (Cambridge University Press, Cambridge, 2018).Google Scholar | DOI

[NT1] Navarro, G. and Tiep, P. H., ‘Rational irreducible characters and rational conjugacy classes in finite groups’, Trans. Amer. Math. Soc. 360 (2008), 2443–2465.Google Scholar | DOI

[NT2] Navarro, G. and Tiep, P. H., ‘Real groups and Sylow 2-subgroups’, Adv. Math. 299 (2016), 331–360.Google Scholar | DOI

[NT3] Navarro, G. and Tiep, P. H., ‘Irreducible representations of odd degree’, Math. Ann. 365 (2016), 1155–1185.Google Scholar | DOI

[NT4] Navarro, G. and Tiep, P. H., ‘Sylow subgroups, exponents, and character tables’, Trans. Amer. Math. Soc. 372 (2019), 4263–4291.Google Scholar | DOI

[NTV] Navarro, G., Tiep, P. H. and Vallejo, C., ‘McKay natural correspondences on characters’, Algebra Number Theory 8 (2014), 1839–1856.Google Scholar | DOI

[O] Olsson, J. B., ‘McKay numbers and heights of characters’, Math. Scand. 38 (1976), 25–42.Google Scholar | DOI

[R] Ruhstorfer, L., ‘The Navarro refinement of the McKay conjecture for finite groups of Lie type in defining characteristic’, Preprint, 2020, .Google Scholar | arXiv

[SF] Schaeffer-Fry, A. A., ‘Actions of Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow -subgroup conjecture’, Trans. Amer. Math. Soc. 372 (2019), 457–483.Google Scholar | DOI

[ST] Schaeffer-Fry, A. A. and Taylor, J., ‘On self-normalising Sylow 2-subgroups in type ’, J. Lie Theory 28 (2018), 139–168.Google Scholar

[Tu] Turull, A., ‘Strengthening the McKay conjecture to include local fields and local Schur indices’, J. Algebra 319 (2008), 4853–4868.Google Scholar | DOI

[TZ] Tiep, P. H. and Zalesskii, A. E., ‘Unipotent elements of finite groups of Lie type and realization fields of their complex representations’, J. Algebra 271 (2004), 327–390.Google Scholar | DOI

[We] Weintraub, S. H., Galois Theory, Universitext (Springer, New York 2009).Google Scholar | DOI

[Wi] Wilson, R. A., ‘The McKay conjecture is true for the sporadic simple groups’, J. Algebra 207 (1998), 294–305.Google Scholar | DOI

Cité par Sources :