This paper by Robert Steinberg discusses the representations of semisimple algebraic groups of characteristic $ p \neq 0 $ and their finite simple groups. The main results include the characterization of irreducible rational projective representations of such groups and their connection to finite simple groups. Steinberg introduces the concept of high weights and shows that every irreducible rational projective representation can be uniquely expressed as a tensor product of certain standard representations. He also proves a conjecture related to the tensor product of representations and establishes that the number of irreducible projective representations of a finite simple algebraic group is $ q^l $, where $ q $ is the number of elements in the rational field and $ l $ is the rank of the group. The paper also discusses the lifting of representations from algebras to groups and the tensor product theorem, which allows the construction of irreducible representations from tensor products of other representations. Steinberg further explores the properties of these representations, including their dimensions and characters, and provides results for specific types of algebraic groups. The paper concludes with a discussion of special isogenies and their implications for the representations of certain algebraic groups. Overall, the paper provides a comprehensive analysis of the representations of semisimple algebraic groups and their finite counterparts in characteristic $ p \neq 0 $.This paper by Robert Steinberg discusses the representations of semisimple algebraic groups of characteristic $ p \neq 0 $ and their finite simple groups. The main results include the characterization of irreducible rational projective representations of such groups and their connection to finite simple groups. Steinberg introduces the concept of high weights and shows that every irreducible rational projective representation can be uniquely expressed as a tensor product of certain standard representations. He also proves a conjecture related to the tensor product of representations and establishes that the number of irreducible projective representations of a finite simple algebraic group is $ q^l $, where $ q $ is the number of elements in the rational field and $ l $ is the rank of the group. The paper also discusses the lifting of representations from algebras to groups and the tensor product theorem, which allows the construction of irreducible representations from tensor products of other representations. Steinberg further explores the properties of these representations, including their dimensions and characters, and provides results for specific types of algebraic groups. The paper concludes with a discussion of special isogenies and their implications for the representations of certain algebraic groups. Overall, the paper provides a comprehensive analysis of the representations of semisimple algebraic groups and their finite counterparts in characteristic $ p \neq 0 $.