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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.


==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 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.


==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>&nbsp;&minus;&nbsp;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]] may be identified with the [[group ring]] of the [[character group]].
*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>&nbsp;&minus;&nbsp;''X''&nbsp;&minus;&nbsp;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''<sub>''F''</sub>(''C''<sub>3</sub>) is isomorphic to '''''Z'''''[''X'',''Y'']/(''X''<sup>2</sup>&nbsp;&minus;&nbsp;''Y''&nbsp;&minus;&nbsp;1, ''XY''&nbsp;&minus;&nbsp;2''Y'',''Y''<sup>2</sup>&nbsp;&minus;&nbsp;3''Y'').
 
*The ring ''R''(S<sup>1</sup>) for the circle group is isomorphic to '''''Z'''''[''X'', ''X''<sup>&nbsp;&minus;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'' &rarr; ''X''<sup>&nbsp;&minus;1</sup>.
 
*The ring ''R''<sub>'''''C'''''</sub>(''S''<sub>3</sub>) for the [[symmetric group]] on three points is isomorphic to '''Z'''[''X'',''Y'']/(''XY''&nbsp;&minus;&nbsp;''Y'',''X''<sup>2</sup>&nbsp;&minus;&nbsp;1,''Y''<sup>2</sup>&nbsp;&minus;&nbsp;''X''&nbsp;&minus;&nbsp;''Y''&nbsp;&minus;&nbsp;1), where ''X'' is the 1-dimensional alternating representation and ''Y'' the 2-dimensional irreducible representation of ''S''<sub>3</sub>.
 
==Characters==
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.
 
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).
 
==Adams operations==
The ''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]]s of ρ and ''N''<sub>''k''</sub> is the ''k''-th [[Newton polynomial]].
 
==References==
*{{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 }}.
*{{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 }}
*{{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 }}.
* {{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 }}
 
[[Category:Group theory]]
[[Category:Finite groups]]
[[Category:Lie groups]]
[[Category:Representation theory of groups]]

Latest revision as of 23:40, 30 July 2014


Any ideas?

Ladies 1st

Many girls feel the pressure of getting to decide on their bestest buddy for the honor of being Matron or Maid in their wedding. They really feel like they are leaving one of thei...

The trick with getting a wedding party is that ultimately youre going to have to decide on somebody to be the best one thing. Girls have it slightly simpler because they can opt to have a Matron of Honor as nicely as a Maid of Honor, but even so, its a challenging selection to make.

Any tips?

Ladies 1st

Numerous ladies feel the pressure of getting to pick their bestest friend for the honor of getting Matron or Maid in their wedding. They feel like they are leaving 1 of their friends out or hurting their feelings. To get another way of interpreting this, consider checking out: clicky. Very first of all, understand that the duty of being in the wedding celebration is massive. The Matron of Honor has a lot to do ahead of and throughout the wedding itself, so you will want to choose an individual that you think can manage it.

If youre the bride, you may possibly really feel that you have to choose a family member as well, but thats not essential. You may possibly contain them as a portion of the celebration, but if you have a truly close pal that you want to bestow the honor upon, then do so.

Find a person who isnt overly busy or committed. If they live in the town that youll be married in, thats fantastic as well. Even if youre out of town, youll get to be in a position to pay a visit to each other and she can make arrangements while youre at your home. You also want to uncover an individual who isnt financially burdened. If you know any thing, you will maybe want to discover about sponsor. The factor is that gowns, showers, bachelorette parties, and gifts all cost income, so you dont want to stretch them.

And then the males

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Selecting a person to be your appropriate hand person is an crucial decision and an honor for the individual that you ask. Though you may feel that they are the correct individual, they may possibly decline the supply and the additional perform. In that case, you will require to discover someone else to fill the part..

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