mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an Adams operation, denoted ψ^''k'' for natural numbers ''k'', is a

cohomology operation In mathematics, the cohomology operation concept became central to algebraic topology, particularly homotopy theory, from the 1950s onwards, in the shape of the simple definition that if ''F'' is a functor defining a cohomology theory, then a coho ...

topological K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...

, or any allied operation in

algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...

or other types of algebraic construction, defined on a pattern introduced by

Frank Adams John Frank Adams (5 November 1930 – 7 January 1989) was a British mathematician, one of the major contributors to homotopy theory. Life He was born in Woolwich, a suburb in south-east London, and attended Bedford School. He began researc ...

. The basic idea is to implement some fundamental identities in

symmetric function In mathematics, a function of n variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f\left(x_1,x_2\right) of two arguments is a symmetric function if and only if f\left(x_1,x_2\right) = f ...

theory, at the level of

vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...

s or other representing object in more abstract theories. Adams operations can be defined more generally in any λ-ring.

Adams operations in K-theory

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

ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...

s. # ψ^''k''(l)= l^k if l is the class of a

line bundle In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the '' tangent bundle'' is a way of organisi ...

. # ψ^''k'' are

functorial In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...

. The fundamental idea is that for a vector bundle ''V'' on a

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...

''X'', there is an analogy between Adams operators and

exterior power In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...

s, in which :ψ^''k''(''V'') is to Λ^''k''(''V'') as :the power sum Σ α^''k'' is to the ''k''-th

elementary symmetric function In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...

σ_''k'' of the roots α of a

polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...

''P''(''t''). (Cf.

Newton's identities In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynom ...

.) Here Λ^''k'' denotes the ''k''-th exterior power. From classical algebra it is known that the power sums are certain integral polynomials ''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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...

). The polynomials here are called Newton polynomials (not, however, the

Newton polynomial In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interp ...

s of

interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has ...

theory). Justification of the expected properties comes from the line bundle case, where ''V'' is a

Whitney sum In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...

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 Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...

(cf. the

Leray–Hirsch theorem In mathematics, the Leray–Hirsch theorem is a basic result on the algebraic topology of fiber bundles. It is named after Jean Leray and Guy Hirsch, who independently proved it in the late 1940s. It can be thought of as a mild generalization of ...

). In general a mechanism for reducing to that case comes from the

splitting principle In mathematics, the splitting principle is a technique used to reduce questions about vector bundles to the case of line bundles. In the theory of vector bundles, one often wishes to simplify computations, say of Chern classes. Often computation ...

for vector bundles.

Adams operations in group representation theory

The Adams operation has a simple expression in

group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used ...

theory. Let ''G'' be a group and ρ a representation of ''G'' with character χ. The representation ψ^''k''(ρ) has character :

\chi_(g) = \chi_\rho(g^k) \ .

References

* {{cite journal , last=Adams , first=J.F. , author-link=Frank Adams , title=Vector Fields on Spheres , journal=

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as th ...

, series=Second Series , volume=75 , number=3 , date=May 1962 , pages=603–632 , zbl=0112.38102 , doi=10.2307/1970213 Algebraic topology Symmetric functions