In
mathematics, especially in the area of
algebra
Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
known as
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
, the representation ring (or Green ring after
J. A. Green) of a
group is a
ring formed from all the (isomorphism classes of the) finite-dimensional linear
representations of the group. Elements of the representation ring are sometimes called virtual representations.
[https://math.berkeley.edu/~teleman/math/RepThry.pdf, page 20] For a given group, the ring will depend on the base field of the representations. The case of complex coefficients is the most developed, but the case of
algebraically closed fields of
characteristic ''p'' where the
Sylow ''p''-subgroups are
cyclic is also theoretically approachable.
Formal definition
Given a group ''G'' and a field ''F'', the elements of its representation ring ''R''
''F''(''G'') are the formal differences of isomorphism classes of finite dimensional linear ''F''-representations of ''G''. For the ring structure, addition is given by the direct sum of representations, and multiplication by their
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), 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 e ...
over ''F''. When ''F'' is omitted from the notation, as in ''R''(''G''), then ''F'' is implicitly taken to be the field of complex numbers.
Succinctly, the representation ring of ''G'' is the
Grothendieck ring
In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in algebraic geometry or number t ...
of the category of finite-dimensional representations of ''G''.
Examples
*For the complex representations of the
cyclic group
In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
of order ''n'', the representation ring ''R''
''C''(''C''
''n'') is isomorphic to Z
'X''(''X''
''n'' − 1), where ''X'' corresponds to the complex representation sending a generator of the group to a primitive ''n''th root of unity.
*More generally, the complex representation ring of a finite
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...
may be identified with the
group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
of the
character group.
*For the rational representations of the cyclic group of order 3, the representation ring ''R''
''Q''(C
3) is isomorphic to ''Z''
'X''(''X''
2 − ''X'' − 2), where ''X'' corresponds to the irreducible rational representation of dimension 2.
*For the modular representations of the cyclic group of order 3 over a field ''F'' of characteristic 3, the representation ring ''R''
''F''(''C''
3) is isomorphic to ''Z''
'X'',''Y''(''X''
2 − ''Y'' − 1, ''XY'' − 2''Y'',''Y''
2 − 3''Y'').
*The continuous representation ring ''R''(S
1) for the circle group is isomorphic to ''Z''
−1">'X'', ''X'' −1 The ring of real representations is the subring of ''R''(''G'') of elements fixed by the involution on ''R''(''G'') given by ''X'' ↦ ''X''
−1.
*The ring ''R''
''C''(''S''
3) for the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
on three points is isomorphic to Z
'X'',''Y''(''XY'' − ''Y'',''X''
2 − 1,''Y''
2 − ''X'' − ''Y'' − 1), where ''X'' is the 1-dimensional alternating representation and ''Y'' the 2-dimensional irreducible representation of ''S''
3.
Characters
Any representation defines a
character
Character or Characters may refer to:
Arts, entertainment, and media Literature
* ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk
* ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
χ:''G'' → C. Such a function is constant on conjugacy classes of ''G'', a so-called
class function
In mathematics, especially in the fields of group theory and representation theory of groups, a class function is a function on a group ''G'' that is constant on the conjugacy classes of ''G''. In other words, it is invariant under the conjug ...
; denote the ring of class functions by ''C''(''G''). If ''G'' is finite, the homomorphism ''R''(''G'') → ''C''(''G'') is injective, so that ''R''(''G'') can be identified with a subring of ''C''(''G''). For fields ''F'' whose characteristic divides the order of the group ''G'', the homomorphism from ''R''
''F''(''G'') → ''C''(''G'') defined by
Brauer characters is no longer injective.
For a compact connected group ''R''(''G'') is isomorphic to the subring of ''R''(''T'') (where ''T'' is a maximal torus) consisting of those class functions that are invariant under the action of the Weyl group (Atiyah and Hirzebruch, 1961). For the general compact Lie group, see Segal (1968).
λ-ring and Adams operations
Given a representation of ''G'' and a natural number ''n'', we can form the ''n''-th
exterior power of the representation, which is again a representation of ''G''. This induces an operation λ
''n'' : ''R''(''G'') → ''R''(''G''). With these operations, ''R''(''G'') becomes a
λ-ring
In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ''n'' on it that behave like the exterior powers of vector spaces. Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide ...
.
The ''
Adams operations In mathematics, an Adams operation, denoted ψ''k'' 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 introd ...
'' on the representation ring ''R''(''G'') are maps Ψ
''k'' characterised by their effect on characters χ:
:
The operations Ψ
''k'' are ring homomorphisms of ''R''(''G'') to itself, and on representations ρ of dimension ''d''
:
where the Λ
''i''ρ are the
exterior powers of ρ and ''N''
''k'' is the ''k''-th power sum expressed as a function of the ''d'' elementary symmetric functions of ''d'' variables.
References
*.
*
*.
* {{citation , 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
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer.
Cambr ...
, year=1994 , isbn=0-521-46015-8 , zbl=0991.20005 , url-access=registration , url=https://archive.org/details/explicitbrauerin0000snai
Group theory
Ring theory
Finite groups
Lie groups
Representation theory of groups