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| In [[mathematics]], especially in the area of [[abstract algebra|algebra]] known as [[representation theory]], the '''representation ring''' (or '''Green ring''' after [[J. A. Green]]) of a [[group (mathematics)|group]] is a [[Ring (mathematics)|ring]] formed from all the (isomorphism classes of the) finite-dimensional linear [[group representation|representations]] of the group. 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 field]]s of [[characteristic of a ring|characteristic]] ''p'' where the [[Sylow subgroup|Sylow ''p''-subgroups]] are [[cyclic group|cyclic]] is also theoretically approachable.
| | I am Alva and was born on 3 February 1970. My hobbies are Cheerleading and Rock stacking.<br><br>my site ... [http://tinyurl.com/nrfwlnd nike air max] |
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| ==Formal definition==
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| 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 linear ''F''-representations of ''G''. For the ring structure, addition is given by the direct sum of representations, and multiplication by their [[tensor product]] 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.
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| ==Examples==
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| *For the complex representations of the [[cyclic group]] of order ''n'', the representation ring ''R''<sub>'''''C'''''</sub>(''C''<sub>''n''</sub>) is isomorphic to '''Z'''[''X'']/(''X''<sup>''n''</sup> − 1), where ''X'' corresponds to the complex representation sending a generator of the group to a primitive ''n''th root of unity.
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| *More generally, the complex representation ring of a finite [[abelian group]] may be identified with the [[group ring]] of the [[character group]].
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| *For the rational 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> − ''X'' − 2), where ''X'' corresponds to the irreducible rational representation of dimension 2.
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| *For the modular representations 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>2</sup> − ''Y'' − 1, ''XY'' − 2''Y'',''Y''<sup>2</sup> − 3''Y'').
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| *The ring ''R''(S<sup>1</sup>) for the circle group is isomorphic to '''''Z'''''[''X'', ''X''<sup> −1</sup>]. The ring of real representations is the subring of ''R''(''G'') of elements fixed by the involution on ''R''(''G'') given by ''X'' → ''X''<sup> −1</sup>.
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| *The ring ''R''<sub>'''''C'''''</sub>(''S''<sub>3</sub>) for the [[symmetric group]] on three points is isomorphic to '''Z'''[''X'',''Y'']/(''XY'' − ''Y'',''X''<sup>2</sup> − 1,''Y''<sup>2</sup> − ''X'' − ''Y'' − 1), where ''X'' is the 1-dimensional alternating representation and ''Y'' the 2-dimensional irreducible representation of ''S''<sub>3</sub>.
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| ==Characters==
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| Any representation defines a [[Character theory|character]] χ:''G'' → '''C'''. Such a function is constant on conjugacy classes of ''G'', a so-called [[class function]]; denote the ring of class functions by ''C''(''G''). 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''<sub>''F''</sub>(''G'') → ''C''(''G'') defined by [[modular representation theory|Brauer characters]] is no longer injective.
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| 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).
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| ==Adams operations==
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| The ''Adams operations'' on the representation ring ''R''(''G'') are maps Ψ<sup>''k''</sup> characterised by their effect on characters χ:
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| :<math>\Psi^k \chi (g) = \chi(g^k) \ . </math>
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| The operations Ψ<sup>''k''</sup> are ring homomorphisms of ''R''(''G'') to itself, and on representations ρ of dimension ''d''
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| :<math>\Psi^k (\rho) = N_k(\Lambda^1\rho,\Lambda^2\rho,\ldots,\Lambda^d\rho) \ </math>
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| where the Λ<sup>''i''</sup>ρ are the [[exterior power]]s of ρ and ''N''<sub>''k''</sub> is the ''k''-th [[Newton polynomial]].
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| ==References==
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| *{{Citation |authorlink1=Michael Atiyah | last1=Atiyah |first1= Michael F. |last2=Hirzebruch| first2=Friedrich | title=Vector bundles and homogeneous spaces |year=1961|journal= Proc. Sympos. Pure Math. | volume=III | publisher=American Mathematical Society | pages=7–38 | mr=0139181 | zbl=0108.17705 }}.
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| *{{Citation | last1=Bröcker| first1=Theodor| last2=tom Dieck | first2=Tammo| title=Representations of Compact Lie Groups | publisher=[[Springer-Verlag]] | location=New York, Berlin, Heidelberg, Tokyo | series=[[Graduate Texts in Mathematics]] | isbn= 0-387-13678-9| mr = 1410059| year=1985 | volume=98 | oclc=11210736 | zbl=0581.22009 }}
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| *{{Citation |last=Segal |first= Graeme |title=The representation ring of a compact Lie group |year=1968|journal=Publ. Math. De l'IHES | volume=34 | pages=113–128 | mr=0248277 | zbl=0209.06203 }}.
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| * {{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 | authorlink= | publisher=[[Cambridge University Press]] | year=1994 | isbn=0-521-46015-8 | zbl=0991.20005 }}
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| [[Category:Group theory]]
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| [[Category:Finite groups]]
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| [[Category:Lie groups]]
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| [[Category:Representation theory of groups]]
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I am Alva and was born on 3 February 1970. My hobbies are Cheerleading and Rock stacking.
my site ... nike air max