Skip to main content Accessibility help
×
Home

Robinson’s conjecture on heights of characters

  • Zhicheng Feng (a1), Conghui Li (a2), Yanjun Liu (a3), Gunter Malle (a4) and Jiping Zhang (a5)...

Abstract

Geoffrey Robinson conjectured in 1996 that the $p$ -part of character degrees in a $p$ -block of a finite group can be bounded in terms of the center of a defect group of the block. We prove this conjecture for all primes $p\neq 2$ for all finite groups. Our argument relies on a reduction by Murai to the case of quasi-simple groups which are then studied using deep results on blocks of finite reductive groups.

Copyright

Footnotes

Hide All

The first and fourth authors gratefully acknowledge financial support by SFB TRR 195, and the others by NSFC (No. 11631001). In addition, the third author gratefully acknowledges financial support by NSFC (No. 11661042) and the Key Laboratory of Mathematics and Its Applications of Peking University.

Footnotes

References

Hide All
[BO97] Bessenrodt, C. and Olsson, J. B., Heights of spin characters in characteristic 2 , in Finite reductive groups, related structures and representations (Birkhäuser, Boston, 1997), 5171.
[BDR17] Bonnafé, C., Dat, J.-F. and Rouquier, R., Derived categories and Deligne–Lusztig varieties II , Ann. of Math. (2) 185 (2017), 609670.
[Bra41] Brauer, R., Investigations on group characters , Ann. of Math. (2) 42 (1941), 936958.
[Bra68] Brauer, R., On blocks and sections in finite groups. II , Amer. J. Math. 90 (1968), 895925.
[BMM93] Broué, M., Malle, G. and Michel, J., Generic blocks of finite reductive groups , Astérisque 212 (1993), 792.
[CE94] Cabanes, M. and Enguehard, M., On unipotent blocks and their ordinary characters , Invent. Math. 117 (1994), 149164.
[CE04] Cabanes, M. and Enguehard, M., Representation theory of finite reductive groups (Cambridge University Press, Cambridge, 2004).
[Car85] Carter, R., Finite groups of Lie type: conjugacy classes and complex characters (Wiley, Chichester, 1985).
[CCNPW84] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A., Atlas of finite groups (Clarendon Press, Oxford, 1984).
[Eat04] Eaton, C. W., The equivalence of some conjectures of Dade and Robinson , J. Algebra 271 (2004), 638651.
[Eng00] Enguehard, M., Sur les l-blocs unipotents des groupes réductifs finis quand l est mauvais , J. Algebra 230 (2000), 334377.
[Eng13] Enguehard, M., Towards a Jordan decomposition of blocks of finite reductive groups, Preprint (2013), arXiv:1312.0106.
[Fon61] Fong, P., On the characters of p-solvable groups , Trans. Amer. Math. Soc. 98 (1961), 263284.
[GAP18] GAP Group, GAP – groups, algorithms, and programming, Version 4.8.10 (2018),http://www.gap-system.org.
[Gec03] Geck, M., On the p-defects of character degrees of finite groups of Lie type , Carpathian J. Math. 19 (2003), 97100.
[GLS98] Gorenstein, D., Lyons, R. and Solomon, R., The classification of the finite simple groups, Mathematical Surveys and Monographs, vol. 40.3 (American Mathematical Society, Providence, RI, 1998).
[KM13] Kessar, R. and Malle, G., Quasi-isolated blocks and Brauer’s height zero conjecture , Ann. of Math. (2) 178 (2013), 321384.
[Lan78] Landrock, P., The non-principal 2-blocks of sporadic simple groups , Comm. Algebra 6 (1978), 18651891.
[Lus84] Lusztig, G., Characters of reductive groups over a finite field, Annals of Mathematics Studies, vol. 107 (Princeton University Press, Princeton, NJ, 1984).
[Mal90] Malle, G., Die unipotenten Charaktere von 2 F 4(q 2) , Comm. Algebra 18 (1990), 23612381.
[Mal91] Malle, G., The maximal subgroups of  2 F 4(q 2) , J. Algebra 139 (1991), 5269.
[Mal07] Malle, G., Height 0 characters of finite groups of Lie type , Represent. Theory 11 (2007), 192220.
[MT11] Malle, G. and Testerman, D., Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133 (Cambridge University Press, Cambridge, 2011).
[Mur98] Murai, M., Blocks of factor groups and heights of characters , Osaka J. Math. 35 (1998), 835854.
[Nav98] Navarro, G., Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250 (Cambridge University Press, Cambridge, 1998).
[Ols93] Olsson, J. B., Combinatorics and representations of finite groups (Universität Essen, Fachbereich Mathematik, Essen, 1993).
[Ree61a] Ree, R., A family of simple groups associated with the simple Lie algebra of type (F 4) , Amer. J. Math. 83 (1961), 401420.
[Ree61b] Ree, R., A family of simple groups associated with the simple Lie algebra of type (G 2) , Amer. J. Math. 83 (1961), 432462.
[Rob96] Robinson, G., Local structure, vertices and Alperin’s conjecture , Proc. Lond. Math. Soc. (3) 72 (1996), 312330.
[Sam14] Sambale, B., Blocks of finite groups and their invariants, Lecture Notes in Mathematics, vol. 2127 (Springer, Cham, 2014).
[Sys80] Syskin, S. A., Abstract properties of the simple sporadic groups , Uspekhi Mat. Nauk 35 (1980), 181212.
[Wag77] Wagner, A., An observation on the degrees of projective representations of the symmetric and alternating group over an arbitrary field , Arch. Math. 29 (1977), 583589.
[Wat78/79] Watanabe, A., On Fong’s reductions , Kumamoto J. Sci. (Math.) 13 (1978/79), 4854.
[Wil09] Wilson, R. A., The finite simple groups, Graduate Texts in Mathematics, vol. 251 (Springer, London, 2009).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Robinson’s conjecture on heights of characters

  • Zhicheng Feng (a1), Conghui Li (a2), Yanjun Liu (a3), Gunter Malle (a4) and Jiping Zhang (a5)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed