# Representation ring

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In [mathematics](/source/mathematics), especially in the area of [algebra](/source/abstract_algebra) known as [representation theory](/source/representation_theory), the '''representation ring''' (or '''Green ring''' after [J. A. Green](/source/Sandy_Green_(mathematician))) of a [group](/source/group_(mathematics)) is a [ring](/source/ring_(mathematics)) formed from all the ([isomorphism class](/source/isomorphism_class)es of the) finite-dimensional [linear representations](/source/group_representation) of the group. Elements of the representation ring are sometimes called virtual representations.<ref>https://math.berkeley.edu/~teleman/math/RepThry.pdf, page 20</ref> For a given group, the ring will depend on the base [field](/source/field_(mathematics)) of the representations.  The case of [complex](/source/complex_number) coefficients is the most developed, but the case of [algebraically closed field](/source/algebraically_closed_field)s of [characteristic](/source/characteristic_(algebra)) ''p'' where the [Sylow ''p''-subgroups](/source/Sylow_subgroup) are [cyclic](/source/cyclic_group) is also theoretically approachable.

==Formal definition==
Given a group ''G'' and a field ''F'', the elements of its '''representation ring''' ''R''<sub>''F''</sub>(''G'') are the formal differences of isomorphism classes of finite-dimensional ''F''-representations of ''G''.  For the ring structure, addition is given by the [direct sum](/source/Direct_sum) of representations, and multiplication by their [tensor product](/source/tensor_product_of_representations) 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.

The representation ring of ''G'' is the [Grothendieck ring](/source/Grothendieck_group) of the [category](/source/category_(mathematics)) of finite-dimensional representations of ''G''.

==Examples==
*For the complex representations of the [cyclic group](/source/cyclic_group) of [order](/source/order_(group_theory)) ''n'', the representation ring ''R''<sub>'''''C'''''</sub>(''C''<sub>''n''</sub>) is [isomorphic](/source/isomorphic) to  '''Z'''[''X'']/(''X''<sup>''n''</sup>&nbsp;−&nbsp;1), where ''X'' corresponds to the complex representation sending a generator of the group to a [primitive](/source/Root_of_unity) ''n''th [root of unity](/source/root_of_unity).
*More generally, the complex representation ring of a [finite](/source/finite_group) [abelian group](/source/abelian_group) may be identified with the [group ring](/source/group_ring) of the [character group](/source/character_group).
*For the [rational](/source/rational_number) representations of the cyclic group of order 3, the representation ring ''R''<sub>'''Q'''</sub>(C<sub>3</sub>) is isomorphic to '''Z'''[''X'']/(''X''<sup>2</sup>&nbsp;−&nbsp;''X''&nbsp;−&nbsp;2), where ''X'' corresponds to the [irreducible](/source/irreducible_representation) rational representation of dimension 2.
*For the [modular representation](/source/modular_representation)s of the cyclic group of order 3 over a field ''F'' of characteristic 3,  the representation ring ''R''<sub>''F''</sub>(''C''<sub>3</sub>) is isomorphic to '''Z'''[''X'',''Y'']/(''X''<sup>&thinsp;2</sup>&nbsp;−&nbsp;''Y''&nbsp;−&nbsp;1, ''XY''&nbsp;−&nbsp;2''Y'',''Y''<sup>&thinsp;2</sup>&nbsp;−&nbsp;3''Y'').
*The continuous representation ring ''R''(S<sup>1</sup>) for the [circle group](/source/circle_group) is isomorphic to '''Z'''[''X'', ''X''<sup>&thinsp;−1</sup>].   The ring of [real](/source/real_number) representations is the [subring](/source/subring) of ''R''(''G'') of elements fixed by the [involution](/source/involution_(mathematics)) on ''R''(''G'') given by ''X'' ↦ ''X''<sup>&thinsp;−1</sup>.
*The ring ''R''<sub>'''C'''</sub>(''S''<sub>3</sub>) for the [symmetric group](/source/symmetric_group) of degree three is isomorphic to '''Z'''[''X'',''Y'']/(''XY''&nbsp;−&nbsp;''Y'',''X''<sup>&thinsp;2</sup>&nbsp;−&nbsp;1,''Y''<sup>&thinsp;2</sup>&nbsp;−&nbsp;''X''&nbsp;−&nbsp;''Y''&nbsp;−&nbsp;1), where ''X'' is the {{nowrap|1-dimensional}} alternating representation and ''Y'' the {{nowrap|2-dimensional}} irreducible representation of ''S''<sub>3</sub>.

==Characters==
Any finite-dimensional complex representation ρ of a group ''G'' defines a function χ:''G'' → <math>\mathbb{C}</math> by the formula χ(''g'') = tr(ρ(''g'')). Such a function is a so-called [class function](/source/class_function), meaning that it is constant on each [conjugacy class](/source/conjugacy_class) of ''G''.  Denote the ring of complex-valued class functions by ''C''(''G'').  The map sending isomorphism classes of representations to their characters gives a [homomorphism](/source/homomorphism) ''R''(''G'') → ''C''(''G''), and when ''G'' is finite this is [injective](/source/injective), so that ''R''(''G'') can be identified with a subring of ''C''(''G'').   

In the case of finite groups this ring homomorphism ''R''(''G'') → ''C''(''G'') extends to an algebra isomorphism  <math> \mathbb{C} \otimes_{\mathbb{Z}} </math> ''R''(''G'') → ''C''(''G'').   Since isomorphism classes of [irreducible representations](/source/irreducible_representations) of a finite group form a basis of <math> \mathbb{C} \otimes_{\mathbb{Z}} </math>''R''(''G''), while [characteristic functions](/source/characteristic_functions) of conjugacy classes form a basis of ''C''(''G''), this shows that a finite group has as many isomorphism classes of irreducible representations as it has conjugacy classes.<ref>{{Cite book |last=Serre |first=Jean-Pierre |title=Linear Representations of Finite Groups |date=1977 |publisher=Springer New York |isbn=978-1-4684-9458-7 |location=New York, NY |oclc=853264255}}</ref>

For a [compact](/source/compact_group) [connected](/source/connected_space) [Lie group](/source/Lie_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](/source/Lie_group), see Segal (1968).

==λ-ring and Adams operations==
Given a representation of ''G'' and a [natural number](/source/natural_number) ''n'', we can form the ''n''-th [exterior power](/source/exterior_power) of the representation, which is again a representation of ''G''. This induces an operation λ<sup>''n''</sup> : ''R''(''G'') → ''R''(''G''). With these operations, ''R''(''G'') becomes a [λ-ring](/source/%CE%BB-ring).

The ''[Adams operations](/source/Adams_operations)'' on the representation ring ''R''(''G'') are maps Ψ<sup>''k''</sup> characterised by their effect on characters χ:
:<math>\Psi^k \chi (g) = \chi(g^k) \ . </math>

The operations Ψ<sup>''k''</sup> are ring homomorphisms of ''R''(''G'') to itself, and on representations ''ρ'' of dimension ''d''

:<math>\Psi^k (\rho) = N_k(\Lambda^1\rho,\Lambda^2\rho,\ldots,\Lambda^d\rho) \ </math>

where the Λ<sup>''i''</sup>''ρ'' are the [exterior power](/source/exterior_power)s of ''ρ'' and ''N''<sub>''k''</sub> is the ''k''-th power sum expressed as a function of the ''d'' elementary symmetric functions of ''d'' variables.

==References==
{{Reflist}}
*{{Citation |author-link1=Michael Atiyah | last1=Atiyah |first1= Michael F. |last2=Hirzebruch| first2=Friedrich |author-link2=Friedrich Hirzebruch | title=Vector bundles and homogeneous spaces |year=1961|journal= Proc. Sympos. Pure Math. | series=Proceedings of Symposia in Pure Mathematics | volume=III | publisher=American Mathematical Society | pages=7–38 | doi=10.1090/pspum/003/0139181 | isbn=9780821814031 | mr=0139181 | zbl=0108.17705 }}.
*{{Citation | last1=Bröcker| first1=Theodor| last2=tom Dieck | first2=Tammo| title=Representations of Compact Lie Groups | publisher=[Springer-Verlag](/source/Springer-Verlag) | location=New York, Berlin, Heidelberg, Tokyo | series=[Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics) | isbn= 0-387-13678-9| mr = 1410059| year=1985 | volume=98 | oclc=11210736 | zbl=0581.22009 }}
*{{Citation |last=Segal |first= Graeme |title=The representation ring of a compact Lie group |year=1968|journal=Publ. Math. IHÉS | volume=34 | pages=113–128 |doi= 10.1007/BF02684592 | mr=0248277 | zbl=0209.06203 |s2cid= 55847918 |url= http://www.numdam.org/item/PMIHES_1968__34__113_0/ |url-access=subscription }}.
* {{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](/source/Cambridge_University_Press) | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 | url-access=registration | url=https://archive.org/details/explicitbrauerin0000snai }}

Category:Group theory
Category:Ring theory
Category:Finite groups
Category:Lie groups
Category:Representation theory of groups

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