Rationality of blocks of quasi-simple finite groups

Farrell, Niamh; Kessar, Radha

doi:10.1090/ert/530

Rationality of blocks of quasi-simple finite groups
HTML articles powered by AMS MathViewer

by Niamh Farrell and Radha Kessar

Represent. Theory 23 (2019), 325-349

DOI: https://doi.org/10.1090/ert/530

Published electronically: September 30, 2019

PDF | Request permission

Abstract:

Let $\ell$ be a prime number. We show that the Morita Frobenius number of an $\ell$-block of a quasi-simple finite group is at most $4$ and that the strong Frobenius number is at most $4 |D|^2!$, where $D$ denotes a defect group of the block. We deduce that a basic algebra of any block of the group algebra of a quasi-simple finite group over an algebraically closed field of characteristic $\ell$ is defined over a field with $\ell ^a$ elements for some $a \leq 4$. We derive consequences for Donovan’s conjecture. In particular, we show that Donovan’s conjecture holds for $\ell$-blocks of special linear groups.

References

J. L. Alperin, Local representation theory, The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979) Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R.I., 1980, pp. 369–375. MR 604606
David Benson and Radha Kessar, Blocks inequivalent to their Frobenius twists, J. Algebra 315 (2007), no. 2, 588–599. MR 2351880, DOI 10.1016/j.jalgebra.2007.03.044
Cédric Bonnafé, Quasi-isolated elements in reductive groups, Comm. Algebra 33 (2005), no. 7, 2315–2337. MR 2153225, DOI 10.1081/AGB-200063602
Cédric Bonnafé, Sur les caractères des groupes réductifs finis à centre non connexe: applications aux groupes spéciaux linéaires et unitaires, Astérisque 306 (2006), vi+165 (French, with English and French summaries). MR 2274998
Cédric Bonnafé, Jean-François Dat, and Raphaël Rouquier, Derived categories and Deligne-Lusztig varieties II, Ann. of Math. (2) 185 (2017), no. 2, 609–670. MR 3612005, DOI 10.4007/annals.2017.185.2.5
Michel Broué, Gunter Malle, and Jean Michel, Generic blocks of finite reductive groups, Astérisque 212 (1993), 7–92. Représentations unipotentes génériques et blocs des groupes réductifs finis. MR 1235832
Marc Cabanes, Local methods for blocks of finite simple groups, Local representation theory and simple groups, EMS Ser. Lect. Math., Eur. Math. Soc., Zürich, 2018, pp. 179–265. MR 3821140
Marc Cabanes and Michel Enguehard, On unipotent blocks and their ordinary characters, Invent. Math. 117 (1994), no. 1, 149–164. MR 1269428, DOI 10.1007/BF01232237
Marc Cabanes and Michel Enguehard, On blocks of finite reductive groups and twisted induction, Adv. Math. 145 (1999), no. 2, 189–229. MR 1704575, DOI 10.1006/aima.1998.1814
Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756, DOI 10.1017/CBO9780511542763
Joseph Chuang and Radha Kessar, On perverse equivalences and rationality, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2018, pp. 257–262. MR 3887770
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841, DOI 10.1017/CBO9781139172417
C. W. Eaton, F. Eisele, and M. Livesey, Donovan’s conjecture, blocks with abelian defect groups and discrete valuation rings, Mathematische Zeitschrift, 2019.
Charles W. Eaton, Radha Kessar, Burkhard Külshammer, and Benjamin Sambale, 2-blocks with abelian defect groups, Adv. Math. 254 (2014), 706–735. MR 3161112, DOI 10.1016/j.aim.2013.12.024
Charles W. Eaton and Michael Livesey, Donovan’s conjecture and blocks with abelian defect groups, Proc. Amer. Math. Soc. 147 (2019), no. 3, 963–970. MR 3896046, DOI 10.1090/proc/14316
C. W. Eaton and M. Livesey, Towards Donovan’s conjecture for abelian defect groups, J. Algebra 519, (2019), 39–61.
Michel Enguehard, Sur les $l$-blocs unipotents des groupes réductifs finis quand $l$ est mauvais, J. Algebra 230 (2000), no. 2, 334–377 (French). MR 1775796, DOI 10.1006/jabr.2000.8318
Niamh Farrell, On the Morita Frobenius numbers of blocks of finite reductive groups, J. Algebra 471 (2017), 299–318. MR 3569187, DOI 10.1016/j.jalgebra.2016.08.043
Farrell, N. Rationality of blocks of quasi-simple finite groups. PhD thesis, City, University of London, 2017.
The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.7.9, 2015.
Meinolf Geck, Character values, Schur indices and character sheaves, Represent. Theory 7 (2003), 19–55. MR 1973366, DOI 10.1090/S1088-4165-03-00170-5
Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer, CHEVIE—a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 175–210. Computational methods in Lie theory (Essen, 1994). MR 1486215, DOI 10.1007/BF01190329
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 4. Part II. Chapters 1–4, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1999. Uniqueness theorems; With errata: The classification of the finite simple groups. Number 3. Part I. Chapter A [Amer. Math. Soc., Providence, RI, 1998; MR1490581 (98j:20011)]. MR 1675976, DOI 10.1090/surv/040.4
G. Hiss, Zerlegungszahlen endlicher Gruppen vom Lie-Typ in nicht-definierender Charakteristik, Habilitationsschrift, 1990.
Gerhard Hiss, On a question of Brauer in modular representation theory of finite groups, Sūrikaisekikenkyūsho K\B{o}kyūroku 1149 (2000), 21–29. Representation theory of finite groups and related topics (Japanese) (Kyoto, 1998). MR 1796360
Gerhard Hiss and Radha Kessar, Scopes reduction and Morita equivalence classes of blocks in finite classical groups. II, J. Algebra 283 (2005), no. 2, 522–563. MR 2111208, DOI 10.1016/j.jalgebra.2004.08.030
Radha Kessar, A remark on Donovan’s conjecture, Arch. Math. (Basel) 82 (2004), no. 5, 391–394. MR 2061445, DOI 10.1007/s00013-004-4880-8
Radha Kessar and Markus Linckelmann, Descent of equivalences and character bijections, Geometric and topological aspects of the representation theory of finite groups, Springer Proc. Math. Stat., vol. 242, Springer, Cham, 2018, pp. 181–212. MR 3901160, DOI 10.1007/978-3-319-94033-5_{7}
Radha Kessar and Gunter Malle, Quasi-isolated blocks and Brauer’s height zero conjecture, Ann. of Math. (2) 178 (2013), no. 1, 321–384. MR 3043583, DOI 10.4007/annals.2013.178.1.6
Markus Linckelmann, The block theory of finite group algebras. Vol. II, London Mathematical Society Student Texts, vol. 92, Cambridge University Press, Cambridge, 2018. MR 3821517
M. Linckelmann, The strong Frobenius numbers for cyclic defect blocks are equal to one, Comm. Algebra (2019), 1–8.
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
Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737, DOI 10.1017/CBO9780511994777
Hirosi Nagao and Yukio Tsushima, Representations of finite groups, Academic Press, Inc., Boston, MA, 1989. Translated from the Japanese. MR 998775
Jeremy Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London Math. Soc. (3) 72 (1996), no. 2, 331–358. MR 1367082, DOI 10.1112/plms/s3-72.2.331
M. Sch\accent127 onert et al, GAP – Groups, Algorithms, and Programming – version 3 release 4 patchlevel 4, Lehrstuhl D f\accent127 ur Mathematik, Rheinisch Westf\accent127 alische Technische Hochschule, Aachen, Germany, 1997.
Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York-Berlin, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237, DOI 10.1007/978-1-4757-5673-9
Jay Taylor, Finding characters satisfying a maximal condition for their unipotent support, J. Pure Appl. Algebra 218 (2014), no. 3, 474–496. MR 3124212, DOI 10.1016/j.jpaa.2013.06.016