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representation ring
In mathematics, especially in the area of algebra known as representation theory, 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. 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 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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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