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Group ring
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=== Representations of a group algebra === Taking ''K''[''G''] to be an abstract algebra, one may ask for [[group representation|representations]] of the algebra acting on a ''K-''vector space ''V'' of dimension ''d''. Such a representation :<math>\tilde{\rho}:K[G]\rightarrow \mbox{End} (V)</math> is an algebra homomorphism from the group algebra to the algebra of [[endomorphism]]s of ''V'', which is isomorphic to the ring of ''d Γ d'' matrices: <math>\mathrm{End}(V)\cong M_{d}(K) </math>. Equivalently, this is a [[module (mathematics)|left ''K''[''G'']-module]] over the abelian group ''V''. Correspondingly, a group representation :<math>\rho:G\rightarrow \mbox{Aut}(V),</math> is a group homomorphism from ''G'' to the group of linear automorphisms of ''V'', which is isomorphic to the [[general linear group]] of invertible matrices: <math>\mathrm{Aut}(V)\cong \mathrm{GL}_d(K) </math>. Any such representation induces an algebra representation :<math>\tilde{\rho}:K[G]\rightarrow \mbox{End}(V),</math> simply by letting <math>\tilde{\rho}(e_g) = \rho(g)</math> and extending linearly. Thus, representations of the group correspond exactly to representations of the algebra, and the two theories are essentially equivalent.
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