×

zbMATH — the first resource for mathematics

Simple endotrivial modules for linear, unitary and exceptional groups. (English) Zbl 1327.20008
The paper under review is a continuation of a paper by C. Lassueur, G. Malle and E. Schulte [J. Reine Angew. Math. (to appear)]. Motivated by a result of G. R. Robinson [Bull. Lond. Math. Soc. 43, No. 4, 712-716 (2011; Zbl 1234.20003)], the aim is to classify the simple endotrivial modules of the quasi-simple finite groups. Here the authors focus their attention on special linear and unitary groups, where they obtain a complete classification, and on exceptional groups of Lie type.
In their earlier paper with Schulte, the authors conjectured that a quasi-simple finite group with a faithful simple endotrivial module in characteristic \(p\) has \(p\)-rank at most \(2\). In the present paper, the authors verify this conjecture. As another consequence of their results, the authors show that the principal \(p\)-block of a finite simple group with noncyclic Sylow \(p\)-subgroup cannot have Loewy length \(4\), for \(p>2\). This result is motivated by a paper by S. Koshitani, B. Külshammer and B. Sambale [Math. Proc. Camb. Philos. Soc. 156, No. 3, 555-570 (2014; Zbl 1329.20009)].

MSC:
20C20 Modular representations and characters
20C33 Representations of finite groups of Lie type
20C34 Representations of sporadic groups
20G05 Representation theory for linear algebraic groups
20G40 Linear algebraic groups over finite fields
Software:
CHEVIE; GAP
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Andrews, GE, \(q\)-analogs of the binomial coefficient congruences of babbage, Wolstenholme and glaisher, Discrete Math., 204, 15-25, (1999) · Zbl 0937.05014
[2] Benson, DJ; Carlson, JF, Nilpotent elements in the Green ring, J. Algebra, 104, 329-350, (1986) · Zbl 0612.20005
[3] Bonnafé, C, Quasi-isolated elements in reductive groups, Comm. Algebra, 33, 2315-2337, (2005) · Zbl 1096.20037
[4] Carlson, J.: Endotrivial modules. In: Recent Developments in Lie Algebras. Groups and Representation Theory, pp. 99-111. Proceeding Symposium of Pure Math., Providence, RI (2012) · Zbl 1216.20004
[5] Carlson, JF; Mazza, N; Nakano, D, Endotrivial modules for the general linear group in nondefining characteristic, Math. Z., 278, 901-925, (2014) · Zbl 1318.20008
[6] Carter, R.: Finite Groups of Lie type: Conjugacy Classes and Complex Characters. Wiley, Chichester (1985) · Zbl 0567.20023
[7] Dudas, O., Malle, G.: Decomposition matrices for exceptional groups at \(d=4\). (Submitted). arXiv:1410.3754 · Zbl 1383.20013
[8] Fong, P; Srinivasan, B, Brauer trees in \(\text{ GL }(n, q)\), Math. Z., 187, 81-88, (1984) · Zbl 0545.20006
[9] Geck, M; Hiss, G; Lübeck, F; Malle, G; Pfeiffer, G, \({\sf CHEVIE}\)—A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, Appl. Algebra Eng. Comm. Comput., 7, 175-210, (1996) · Zbl 0847.20006
[10] Geck, M., Jacon, N.: Representations of Hecke Algebras at Roots of Unity. Algebra and Applications, 15. Springer, London (2011) · Zbl 1232.20008
[11] James, G; Mathas, A, A \(q\)-analogue of the jantzen-schaper theorem, Proc. Lond. Math. Soc., 74, 241-274, (1997) · Zbl 0869.20004
[12] Koshitani, S; Külshammer, B; Sambale, B, On loewy lengths of blocks, Math. Proc. Cambridge Philos. Soc., 156, 555-570, (2014) · Zbl 1329.20009
[13] Koshitani, S; Kunugi, N, The principal 3-blocks of the 3-dimensional projective special unitary groups in non-defining characteristic, J. Reine Angew. Math., 539, 1-27, (2001) · Zbl 1003.20011
[14] Kunugi, N, Morita equivalent 3-blocks of the 3-dimensional projective special linear groups, Proc. Lond. Math. Soc., 80, 575-589, (2000) · Zbl 1029.20005
[15] Landrock, P.: Finite Group Algebras and Their Modules. London Math. Soc. Lecture Note Series, 80. Cambridge University Press, Cambridge (1983) · Zbl 0523.20001
[16] Lassueur, C., Malle, G., Schulte, E.: Simple endotrivial modules for quasi-simple groups. J. Reine Angew. Math. (to appear) doi:10.1515/crelle-2013-0100 · Zbl 1397.20022
[17] Lübeck, F., Malle, G.: A Murnaghan-Nakayama rule for values of unipotent characters in classical groups (2015, preprint) · Zbl 1216.20004
[18] Lusztig, G, A unipotent support for irreducible representations, Adv. Math., 94, 139-179, (1992) · Zbl 0789.20042
[19] Malle, G., Testerman, D.: Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics, 133. Cambridge University Press, Cambridge (2011) · Zbl 1256.20045
[20] Manolov, P, Brauer trees in finite special linear groups, C. R. Acad. Bulgare Sci., 63, 327-330, (2010) · Zbl 1216.20004
[21] Simpson, WA; Frame, JS, The character tables for \(\text{ SL }(3, q)\), \(\text{ SU }(3, q^2)\), \(\text{ PSL }(3, q)\), \(\text{ PSU }(3, q^2)\), Can. J. Math., 25, 486-494, (1973) · Zbl 0264.20010
[22] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.4 (2004), http://www.gap-system.org
[23] Thévenaz, J.: Endo-permutation modules, a guided tour. In: Group Representation Theory, pp. 115-147. EPFL Press, Lausanne (2007) · Zbl 1154.20004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.