5 - Groups in Class J: defining characteristic

Summary

The theory of representations of finite simple groups of Lie type in defining characteristic is somewhat advanced. The representations arise from those of the associated algebraic groups, and so some familiarity with the theory of algebraic groups is necessary in order to understand it. For an introduction to this theory see, for example, the survey article by Humphreys [51]. The enthusiastic reader may wish to consult Jantzen [58] for a more detailed exposition. Humphrey's classic book [50] provide a general exposition of the theory of algebraic groups and their representations, whilst Malle and Testerman's book [91] gives an excellent introduction to the general theory, subgroup structure, and representation theory of the finite and algebraic groups of Lie type, including a fuller discussion of all of the introductory material in this chapter.

In many respects, the study of the J₂-candidates is easier than that of the J₁-candidates, simply because there are far fewer of them: we just need to know about the representations in dimensions up to 12, and to be able to determine some of their properties, such as forms preserved and their behaviour under the actions of group and field automorphisms. Fortunately it is possible to extract this information starting from a superficial familiarity with the main results of the theory, principally the Steinberg Tensor Product Theorems. These theorems, together with the tables in [84], suffice to determine the representations.