groups For a split reductive group G over a field k, the irreducible representations of G (as an algebraic group) are parametrized by the dominant weights, which are defined as the intersection of the weight lattice X(T) ≅ Zn with a convex cone (a Weyl chamber) in Rn. In particular, this parametrization is independent of the characteristic of k. In more detail, fix a split maximal torus and a Borel subgroup, T ⊂ B ⊂ G. Then B is the semidirect product of T with a smooth connected unipotent subgroup U. Define a highest weight vector in a representation V of G over k to be a nonzero vector v such that B maps the line spanned by v into itself. Then B acts on that line through its quotient group T, by some element λ of the weight lattice X(T). Chevalley showed that every irreducible representation of G has a unique highest weight vector up to scalars; the corresponding "highest weight" λ is dominant; and every dominant weight λ is the highest weight of a unique irreducible representation L(λ) of G, up to isomorphism. There remains the problem of describing the irreducible representation with given highest weight. For k of characteristic zero, there are essentially complete answers. For a dominant weight λ, define the Schur module ∇(λ) as the k-vector space of sections of the G-equivariant line bundle on the flag manifold G/B associated to λ; this is a representation of G. For k of characteristic zero, the Borel–Weil theorem says that the irreducible representation L(λ) is isomorphic to the Schur module ∇(λ). Furthermore, the Weyl character formula gives the character (and in particular the dimension) of this representation. For a split reductive group G over a field k of positive characteristic, the situation is far more subtle, because representations of G are typically not direct sums of irreducibles. For a dominant weight λ, the irreducible representation L(λ) is the unique simple submodule (the socle) of the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the Weyl character formula (as in characteristic zero), by George Kempf. The dimensions and characters of the irreducible representations L(λ) are in general unknown, although a large body of theory has been developed to analyze these representations. One important result is that the dimension and character of L(λ)