# Adams operation

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In [mathematics](/source/Mathematics), an **Adams operation**, denoted ψ*k* for natural numbers *k*, is a [cohomology operation](/source/Cohomology_operation) in [topological K-theory](/source/Topological_K-theory), or any allied operation in [algebraic K-theory](/source/Algebraic_K-theory) or other types of algebraic construction, defined on a pattern introduced by [Frank Adams](/source/Frank_Adams). The basic idea is to implement some fundamental identities in [symmetric function](/source/Symmetric_function) theory, at the level of [vector bundles](/source/Vector_bundle) or other representing object in more abstract theories.

Adams operations can be defined more generally in any [λ-ring](/source/Lambda_ring).

## Adams operations in K-theory

Adams operations ψ*k* on K theory (algebraic or topological) are characterized by the following properties.

1. ψ*k* are [ring homomorphisms](/source/Ring_homomorphism).
1. ψ*k*(l)= lk if l is the class of a [line bundle](/source/Line_bundle).
1. ψ*k* are [functorial](/source/Functorial).

The fundamental idea is that for a vector bundle *V* on a [topological space](/source/Topological_space) *X*, there is an analogy between Adams operators and [exterior powers](/source/Exterior_power), in which

- ψ*k*(*V*) is to Λ*k*(*V*)

as

- the [power sum](/source/Power_sum_symmetric_polynomial) Σ α*k* is to the *k*-th [elementary symmetric function](/source/Elementary_symmetric_function) σ*k*

of the roots α of a [polynomial](/source/Polynomial) *P*(*t*). (Cf. [Newton's identities](/source/Newton's_identities).) Here Λ*k* denotes the *k*-th exterior power. From classical algebra it is known that the power sums are certain [integral polynomials](/source/Integral_polynomial) *Q**k* in the σ*k*. The idea is to apply the same polynomials to the Λ*k*(*V*), taking the place of σ*k*. This calculation can be defined in a *K*-group, in which vector bundles may be formally combined by addition, subtraction and multiplication ([tensor product](/source/Tensor_product)). The polynomials here are called **Newton polynomials** (not, however, the [Newton polynomials](/source/Newton_polynomial) of [interpolation](/source/Interpolation) theory).

Justification of the expected properties comes from the line bundle case, where *V* is a [Whitney sum](/source/Whitney_sum) of line bundles. In this special case the result of any Adams operation is naturally a vector bundle, not a linear combination of ones in *K*-theory. Treating the line bundle direct factors formally as roots is something rather standard in [algebraic topology](/source/Algebraic_topology) (cf. the [Leray–Hirsch theorem](/source/Leray%E2%80%93Hirsch_theorem)). In general a mechanism for reducing to that case comes from the [splitting principle](/source/Splitting_principle) for vector bundles.

## Adams operations in group representation theory

The Adams operation has a simple expression in [group representation](/source/Group_representation) theory.[1] Let *G* be a group and ρ a representation of *G* with character χ. The representation ψ*k*(ρ) has character

- \chi_{\psi^k(\rho)}(g) = \chi_\rho(g^k) \ .

## References

1. Snaith, V. P. (1994). [*Explicit Brauer Induction: With Applications to Algebra and Number Theory*](https://archive.org/details/explicitbrauerin0000snai/page/108). Vol. 40. Cambridge Studies in Advanced Mathematics. [Cambridge University Press](/source/Cambridge_University_Press). p. [108](https://archive.org/details/explicitbrauerin0000snai/page/108). ISBN 0-521-46015-8. Zbl 0991.20005.

- Adams, J.F. (May 1962). "Vector Fields on Spheres". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **75** (3): 603–632. Second Series. [doi:10.2307/1970213](https://doi.org/10.2307/1970213). [JSTOR 1970213](https://www.jstor.org/stable/1970213). Zbl 0112.38102.

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