Many rings considered in K-theory carry a natural λ-ring structure.
λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results (Lascoux (2003)).
If V and W are finite-dimensional vector spaces over a field k, then we can form the direct sum V ⊕ W, the tensor product V ⊗ W, and the n-th exterior power of V, Λn(V).
All of these are again finite-dimensional vector spaces over k. The same three operations of direct sum, tensor product and exterior power are also available when working with k-linear representations of a finite group, when working with vector bundles over some topological space, and in more general situations.
If we have a short exact sequence of vector bundles over a smooth scheme
(which is actually a ring), we get this local equation globally for free, from the defining equivalence relations.
So demonstrating the basic relation in a λ-ring,[1] that A λ-ring is a commutative ring R together with operations λn : R → R for every non-negative integer n. These operations are required to have the following properties valid for all x, y in R and all n, m ≥ 0: where Pn and Pn,m are certain universal polynomials with integer coefficients that describe the behavior of exterior powers on tensor products and under composition.
Then Pn is the unique polynomial in 2n variables with integer coefficients such that Pn(e1, ..., en, f1, ..., fn) is the coefficient of tn in the expression The λ-rings defined above are called "special λ-rings" by some authors, who use the term "λ-ring" for a more general concept where the conditions on λn(1), λn(xy) and λm(λn(x)) are dropped.