Abstract
This chapter discusses the theory of Lambda-Rings. The theory of lambda-rings (λ-rings) originates in the work of Grothendieck on the theory of Chern classes. The definition of a λ-ring is obtained by a suitable axiomatization of the algebraic properties of exterior powers operations when they act on vector spaces, vector bundles, and linear representations of groups. Using the λ-ring formalism, Grothendieck constructed universal Chern classes as maps from λ-rings to naturally associated graded rings. The theory of λ-rings appears as a fundamental tool in four fields of mathematics: K-theory, representations of groups, general algebra and algebraic combinatorics, and geometry of convex polytopes. The general definitions and prove or survey of the fundamental results is presented. This includes the definitions of λ-rings, γ-rings and Ψ-rings which are formulated in modern terms. An account of developments of the theory is also provided. The combinatorial λ-ring structures appearing in the symmetric group algebras and in the theory of convex polytopes and toric varieties are also described.
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