Book contents
- Frontmatter
- Contents
- Introduction
- A speech in honour of John Cannon and Derek Holt
- Finite groups of Lie type and their representations
- Iterated monodromy groups
- Engel elements in groups
- Some classes of finite semigroups with kite-like egg-boxes of D-classes
- Structure of finite groups having few conjugacy class sizes
- Group theory in cryptography
- A survey of recent results in groups and orderings: word problems, embeddings and amalgamations
- A survey on the minimum genus and maximum order problems for bordered Klein surfaces
- On one-relator quotients of the modular group
- Miscellaneous results on supersolvable groups
- Automorphisms of products of finite groups
- A rational property of the irreducible characters of a finite group
- Automotives
- On n-abelian groups and their generalizations
- Computing with matrix groups over infinite fields
- Trends in infinite dimensional linear groups
- Engel conditions on orderable groups and in combinatorial problems (a survey)
- References
Finite groups of Lie type and their representations
Published online by Cambridge University Press: 05 July 2011
Book contents
- Frontmatter
- Contents
- Introduction
- A speech in honour of John Cannon and Derek Holt
- Finite groups of Lie type and their representations
- Iterated monodromy groups
- Engel elements in groups
- Some classes of finite semigroups with kite-like egg-boxes of D-classes
- Structure of finite groups having few conjugacy class sizes
- Group theory in cryptography
- A survey of recent results in groups and orderings: word problems, embeddings and amalgamations
- A survey on the minimum genus and maximum order problems for bordered Klein surfaces
- On one-relator quotients of the modular group
- Miscellaneous results on supersolvable groups
- Automorphisms of products of finite groups
- A rational property of the irreducible characters of a finite group
- Automotives
- On n-abelian groups and their generalizations
- Computing with matrix groups over infinite fields
- Trends in infinite dimensional linear groups
- Engel conditions on orderable groups and in combinatorial problems (a survey)
- References
Summary
This article is a slightly expanded account of the series of four lectures I gave at the conference. It is intended as a (non-comprehensive) survey covering some important aspects of the representation theory of finite groups of Lie type, where the emphasis is put on the problem of labelling the irreducible representations and of finding their degrees. All three cases are covered, representations in characteristic zero, in defining as well as in non-defining characteristics.
The first section introduces various ways of defining groups of Lie type and some classes of important subgroups of them. The next three sections are devoted to the representation theory of these groups, each section covering one of the three cases.
The lectures were addressed at a broad audience. Thus on the one hand, I have tried to introduce even the most fundamental notions, but on the other hand, I have also tried to get right to the edge of today's knowledge in the topics discussed. As a consequence, the lectures were of a somewhat inhomogeneous level of difficulty. In this article I have omitted the most introductory material. The reader may find all background material needed from representation theory in the textbook [51] by Isaacs.
For this survey I have included a few more examples, as well as most of the references to the results presented in my talks.
- Type
- Chapter
- Information
- Groups St Andrews 2009 in Bath , pp. 1 - 40Publisher: Cambridge University PressPrint publication year: 2011
References
[1] and , Restricting the Steinberg character in finite symplectic groups, J. Group Theory 9 (2006), 251–264.Google Scholar
[2] , and , Representations of quantum groups at a pth root of unity and of semisimple groups in characteristic p: independence of p, Astérisque No. 220 (1994), 321 pp.Google Scholar
[3] , On the decomposition numbers of the Hecke algebra of G(m, 1, n), J. Math. Kyoto Univ. 36 (1996), 789–808.Google Scholar
[4] , and A geometric setting for the quantum deformation of GLn, Duke Math. J. 61 (1990), 655–677.Google Scholar
[5] and , Catégories dérivées et variétés de Deligne-Lusztig, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 1–59.Google Scholar
[7] , and , Quantum linear groups and representations of GLn (Fq), Mem. Amer. Math. Soc. 149 (2001), no. 706.Google Scholar
[10] , Finite groups of Lie type: Conjugacy classes and complex characters, (John Wiley & Sons, Inc., New York 1985).Google Scholar
[12] , 1956–1958. Classification des groupes de Lie algébriques, 2 vols., (Secrétariat mathématique, 11 rue Pierre Curie, Paris 1958), ii+166 + ii+122 pp.Google Scholar
[14] and , Representations of reductive groups over finite fields, Ann. of Math. 103 (1976), 103–161.Google Scholar
[15] and , Representations of finite groups of Lie type, London Math. Soc. Student Texts 21, (CUP, Cambridge 1991).Google Scholar
[16] , On the decomposition numbers of the finite general linear groups, Trans. Amer. Math. Soc. 290 (1985), 315–344.Google Scholar
[17] , On the decomposition numbers of the finite general linear groups, II, Trans. Amer. Math. Soc. 292 (1985), 123–133.Google Scholar
[19] and , Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20–52.Google Scholar
[20] and , Identification of the irreducible modular representations of GLn(q), J. Algebra 104 (1986), 266–288.Google Scholar
[22] , , and , Representations of Hecke algebras and finite groups of Lie type, in Algorithmic algebra and number theory (Heidelberg, 1997) ( et al., eds.), (Springer-Verlag, Berlin 1999), 331–378.Google Scholar
[23] , A note on quantized Weyl reciprocity at roots of unity, Algebra Colloq. 2 (1995), 363–372.Google Scholar
[24] , Decomposition numbers for symmetric groups and composition factors of Weyl modules, J. Algebra 180 (1996), 316–320.Google Scholar
[25] and , The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), 109–153.Google Scholar
[27] and , The blocks of finite classical groups, J. Reine Angew. Math. 396 (1989), 122–191.Google Scholar
[29] , On the decomposition numbers of the finite unitary groups in non-defining characteristic, Math. Z. 207 (1991), 83–89.Google Scholar
[30] , Generalized Gelfand-Graev characters for Steinberg's triality groups and their applications, Comm. Algebra 19 (1991), 3249–3269.Google Scholar
[32] , Basic sets of Brauer characters of finite groups of Lie type, II, J. London Math. Soc. 47 (1993), 255–268.Google Scholar
[33] and , Basic sets of Brauer characters of finite groups of Lie type, J. Reine Angew. Math. 418 (1991), 173–188.Google Scholar
[34] and , Modular representations of finite groups of Lie type in nondefining characteristic, in Finite reductive groups (Luminy, 1994) (, ed.), Progr. Math., 141, (Birkhäuser Boston, Boston, MA 1997), 195–249.Google Scholar
[35] and , Centers and simple modules for Iwahori-Hecke algebras, in Finite reductive groups (Luminy, 1994) (, ed.), Progr. Math., 141, (Birkhäuser Boston, Boston, MA 1997), 251–272.Google Scholar
[36] , and , Towards a classification of the irreducible representations in non-defining characteristic of a finite group of Lie type, Math. Z. 221 (1996), 353–386.Google Scholar
[37] , The characters of the finite general linear groups, Trans. Amer. Math. Soc. 80 (1955), 402–447.Google Scholar
[38] , Polynomial representations of GLn, Lecture Notes in Mathematics 830, (Springer-Verlag, Berlin-New York 1980).Google Scholar
[39] and , Decomposition numbers of finite classical groups for linear primes, J. Reine Angew. Math. 485 (1997), 55–91.Google Scholar
[40] , Eisenstein series over finite fields, in Functional analysis and related fields, (Springer-Verlag, New York 1970), 76–88.Google Scholar
[41] , On the 2-decomposition numbers of Steinberg's triality groups 3D4(q), q odd, J. Algebra 309 (2007), 569–593.Google Scholar
[42] and , Character table of a Borel subgroup of the Ree groups 2F4(q2), LMS J. Comput. Math. 12 (2009), 1–53.Google Scholar
[44] and , 3-blocks and 3-modular characters of G2(q), J. Algebra 131 (1990), 371–387.Google Scholar
[45] and , 2-blocks and 2-modular characters of the Chevalley groups G2(q), Math. Comp. 59 (1992), 645–672.Google Scholar
[46] and , Induced cuspidal representations and generalized Hecke rings, Invent. Math. 58 (1980), 37–64.Google Scholar
[47] and , On Harish-Chandra induction for modules of Levi subgroups, J. Algebra 165 (1994), 172–183.Google Scholar
[48] , Modular representations of classical Lie algebras and semi-simple groups, J. Algebra 19 (1971), 57–79.Google Scholar
[49] , Linear algebraic groups, Graduate Texts in Mathematics, No. 21, (Springer-Verlag, New York-Heidelberg 1975).Google Scholar
[51] , Character theory of finite groups, Corrected reprint of the 1976 original [Academic Press, New York], (AMS Chelsea Publishing, Providence, RI, 2006).Google Scholar
[52] , On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac. Sci. Univ. Tokyo Sect. I 10 (1964), 215–236.Google Scholar
[53] , The decomposition of tensors over fields of prime characteristic, Math. Z. 172 (1980), 161–178.Google Scholar
[54] , The decomposition matrices of GLn (q) for n ≤ 10, Proc. London Math. Soc. 60 (1990), 225–265.Google Scholar
[55] , , and , An Atlas of Brauer Characters, (Oxford University Press, New York 1995).Google Scholar
[56] , Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren, Math. Z. 140 (1974), 127–149.Google Scholar
[57] , Representations of algebraic groups, Second edition, (AMS, Providence, RI 2003).Google Scholar
[58] and , Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), 165–184.Google Scholar
[59] , Unipotente Charaktere und Zerlegungszahlen der endlichen Chevalleygruppen vom Typ F4, Dissertation, RWTH Aachen University, 2006.
[61] , and , Une conjecture pour le calcul des matrices de décomposition des algèbres de Hecke du type A aux racines de l'unité, C. R. Acad. Sci. Paris Sér. I 321 (1995), 511–516.Google Scholar
[62] , and , Canonical bases of q-deformed Fock spaces, Internat. Math. Res. Notices 1996, no. 9, 447–456.Google Scholar
[63] , Irreducible representations of finite classical groups, Invent. Math. 43 (1977), 125–175.Google Scholar
[64] , Some problems in the representation theory of finite Chevalley groups, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), Proc. Sympos. Pure Math., 37, (Amer. Math. Soc., Providence, R.I. 1980), 313–317.Google Scholar
[65] , Characters of reductive groups over a finite field, Ann. Math. Studies 107, (Princeton University Press, Princeton, NJ 1984).Google Scholar
[66] , On the representations of reductive groups with disconnected centre, in: Orbites unipotentes et représentations, I. Groupes finis et algèbres de Hecke, Astérisque 168 (1988), 157–166.Google Scholar
[67] , Rouquier blocks in Chevalley groups of type E, Adv. Math. 217 (2008), 2841–2871.Google Scholar
[70] , A family of simple groups associated with the simple Lie algebra of type (F4), Amer. J. Math. 83 (1961), 401–420.Google Scholar
[71] , A family of simple groups associated with the simple Lie algebra of type (G2), Amer. J. Math. 83 (1961), 432–462.Google Scholar
[74] , A geometric approach to the representations of the full linear group over a Galois field, Trans. Amer. Math. Soc. 71 (1951), 274–282.Google Scholar
[77] , Lectures on Chevalley groups, Notes prepared by and , (Yale University, New Haven, Conn. 1968).Google Scholar
[79] , A new type of simple groups of finite order, Proc. Nat. Acad. Sci. U.S.A. 46 (1960), 868–870.Google Scholar
[80] , Les “formes réelles” des groupes de type E6, Séminaire Bourbaki; 10e année: 1957/1958. Textes des conférences; Exposés 152 à 168; 2e èd. corrigée, Exposé 162, 15 pp. (Secrétariat mathématique, Paris 1958), 189 pp.
[81] and , On the decomposition matrices of the quantized Schur algebra, Duke Math. J. 100 (1999), 267–297.Google Scholar
[82] , The 2-decomposition numbers of Sp(4, q), q odd, J. Algebra 131 (1990), 703–725.Google Scholar
[83] , Decomposition numbers of Sp(4, q) for primes dividing q±1, J. Algebra 132 (1990), 488–500.Google Scholar
[84] , Decomposition numbers of Sp4(2a) in odd characteristics, J. Algebra 177 (1995), 264–276.Google Scholar
[85] , Decomposition numbers of unipotent blocks of Sp6(2a) in odd characteristics, J. Algebra 227 (2000), 172–194.Google Scholar