In this note, we give the dimensions of some irreducible representations of exceptional Lie algebras and algebraic groups. Similar results appear in [1] for classical groups and algebras of rank at most 4. These results were produced by computer programs developed in connection with [3], where the main result required information beyond the tables in [1]. In view of the utility of the tables in [1], it seemed worthwhile to provide tables for groups of higher rank.
Although our methods are similar to those of [1], they incorporate a reduction process which permits us to push the techniques a bit further.
Let G be a simple algebraic group over an algebraically closed field K, let L~---L(G) be the Lie algebra of G, and let V be an irreducible, finite dimensional, rational representation of G. Then V~-~V(X) where ), is the high weight of V with respect to a fixed maximal torus T of G. Let X-----E el), i where the )'i are the fundamental dominant weights.
Of particular interest is the ease where char(K)----p>0. Here ), is said to be restricted if ci< p for each i. The Steinberg tensor product theorem shows V is the tensor product of twists of restricted modules, so we henceforth assume V is restricted. Then V is also irreducible as a module for L.
The module V has a T-weight space decomposition V=@Vq where ~/ranges over the weights less than ), in the usual ordering of the weights. Each V.~ is