Editing Adams operation

In [[mathematics]], an '''Adams operation''', denoted ψ<sup>''k''</sup> for natural numbers ''k'',  is a [[cohomology operation]] in [[topological K-theory]], or any allied operation in [[algebraic K-theory]] or other types of algebraic construction, defined on a pattern introduced by [[Frank Adams]]. The basic idea is to implement some fundamental identities in [[symmetric function]] theory, at the level of [[vector bundle]]s or other representing object in more abstract theories.

Adams operations can be defined more generally in any [[lambda ring|&lambda;-ring]].

==Adams operations in K-theory==
Adams operations ψ<sup>''k''</sup> on K theory (algebraic or topological) are characterized by the following properties.
# ψ<sup>''k''</sup> are [[ring homomorphism]]s.
# ψ<sup>''k''</sup>(l)= l<sup>k</sup> if l is the class of a [[line bundle]].
# ψ<sup>''k''</sup> are [[functorial]].

The fundamental idea is that for a vector bundle ''V'' on a [[topological space]] ''X'', there is an analogy between Adams operators and [[exterior power]]s, in which

:ψ<sup>''k''</sup>(''V'') is to Λ<sup>''k''</sup>(''V'')

as

:the [[Power sum symmetric polynomial|power sum]] Σ α<sup>''k''</sup> is to the ''k''-th [[elementary symmetric function]] σ<sub>''k''</sub>

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

Justification of the expected properties comes from the line bundle case, where ''V'' is a [[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]] (cf. the [[Leray–Hirsch theorem]]). In general a mechanism for reducing to that case comes from the [[splitting principle]] for vector bundles.

==Adams operations in group representation theory==
The Adams operation has a simple expression in [[group representation]] theory.<ref name=Sn108>{{cite book | title=Explicit Brauer Induction: With Applications to Algebra and Number Theory | volume=40 | series=Cambridge Studies in Advanced Mathematics | first=V. P. | last=Snaith | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 | page=[https://archive.org/details/explicitbrauerin0000snai/page/108 108] | url=https://archive.org/details/explicitbrauerin0000snai/page/108 }}</ref>  Let ''G'' be a group and ρ a representation of ''G'' with character χ.  The representation ψ<sup>''k''</sup>(ρ) has character

:<math>\chi_{\psi^k(\rho)}(g) = \chi_\rho(g^k) \ . </math>

==References==
{{reflist}}
*  {{cite journal | last=Adams | first=J.F. | author-link=Frank Adams | title=Vector Fields on Spheres | journal=[[Annals of Mathematics]] |series=Second Series | volume=75 | number=3 | date=May 1962 | pages=603–632 | zbl=0112.38102 | doi=10.2307/1970213| jstor=1970213 }}

[[Category:Algebraic topology]]
[[Category:Symmetric functions]]