Examples
- For the complex representations of the cyclic group of order n, the representation ring RC(Cn) is isomorphic to Z/(Xn − 1), where X corresponds to the complex representation sending a generator of the group to a primitive nth 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 RQ(C3) is isomorphic to Z/(X2 − 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 RF(C3) is isomorphic to Z/(X2 − Y − 1, XY − 2Y,Y2 − 3Y).
- The ring R(S1) for the circle group is isomorphic to Z. 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 RC(S3) for the symmetric group on three points is isomorphic to Z/(XY − Y,X2 − 1,Y2 − X − Y − 1), where X is the 1-dimensional alternating representation and Y the 2-dimensional irreducible representation of S3.
Read more about this topic: Representation Ring
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