Modular Representations of Finite Groups of Lie Type (London Mathematical Society Lecture Note Series, Series Number 326)

Finite Groups Of Lie Type Encompass Most Of The Finite Simple Groups. Their Representations And Characters Have Been Studied Intensively For Half A Century, Though Some Key Problems Remain Unsolved. This Is The First Comprehensive Treatment Of The Representation Theory Of Finite Groups Of Lie Type Over A Field Of The Defining Prime Characteristic. As A Subtheme, The Relationship Between Ordinary And Modular Representations Is Explored, In The Context Of Deligne–lusztig Characters. One Goal Has Been To Make The Subject More Accessible To Those Working In Neighbouring Parts Of Group Theory, Number Theory, And Topology. Core Material Is Treated In Detail, But The Later Chapters Emphasize Informal Exposition Accompanied By Examples And Precise References. 1. Finite Groups Of Lie Type -- 2. Simple Modules -- 3. Weyl Modules And Lusztig's Conjecture -- 4. Computation Of Weight Multiplicities -- 5. Other Aspects Of Simple Modules -- 6. Tensor Products -- 7. Bn-pairs And Induced Modules -- 8. Blocks -- 9. Projective Modules -- 10. Comparison With Frobenius Kernels -- 11. Cartan Invariants -- 12. Extensions Of Simple Modules -- 13. Loewy Series -- 14. Cohomology -- 15. Complexity And Support Varieties -- 16. Ordinary And Modular Representations -- 17. Deligne-lusztig Characters -- 18. Groups G[subscript 2](q) -- 19. General And Special Linear Groups -- 20. Suzuki And Ree Groups. James E. Humphreys. Includes Bibliographical References (p. 213-228) And Index. Cover; Title; Copyright; Contents; Preface; 1 Finite Groups of Lie Type; 1.1 Algebraic Groups over Finite Fields; 1.2 Classifi cation Over Finite Fields; 1.3 Frobenius Maps; 1.4 Lang Maps; 1.5 Chevalley Groups and Twisted Groups; 1.6 Example: SL(3, q) and SU(3, q); 1.7 Groups With a BN-Pair; 1.8 Notational Conventions; 2 Simple Modules; 2.1 Representations and Formal Characters; 2.2 Simple Modules for Algebraic Groups; 2.3 Construction of Modules; 2.4 Contravariant Forms; 2.5 Representations of Frobenius Kernels; 2.6 Invariants in the Function Algebra; 2.7 Steinberg's Tensor Product Theorem We begin with a brief review of the standard ways in which finite groups of Lie type are classified, constructed and described.