Theory of dimensions, finite and infinite by Ryszard Engelking

By Ryszard Engelking

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Given a representation ρ of G, its representation space V may be viewed as a G-module, where G acts on V by xv = ρ(x)(v), for x ∈ G and v ∈ V . Conversely, a Gmodule V is the representation space of a representation ρ by setting ρ(x)(v) = xv, x ∈ G and v ∈ V . This correspondence enables us to view each representation of a group G as a G-module, and conversely. If V is a G-module with corresponding representation ρ, the representation of G that corresponds to any sub G-module W of the G-module V is called a subrepresentation of ρ.

Iii) The semisimple representation ρ associated with ρ is trivial. Proof. (i)⇒ (ii) is trivial. 1) such that the induced action of G on each factor Vi+1 /Vi is trivial. It follows from this that the semisimple representation ρ associated with ρ is trivial. 1) of subspaces of V for which the induced action of G on each factor Vi+1 /Vi is trivial. Therefore, for any x ∈ G, we have (ρ(x) − 1)(Vi ) ⊂ Vi−1 , 1 ≤ i ≤ n, where V0 = (0), and hence (i) follows. Closed Sets of Representations Let G be a complex (resp.

In the case G is an analytic group having the property that every analytic representation of G is semisimple, then we may construct [E] explicitly from E as follows. Let E1 denote the set of all irreducible subrepresentations of all finite tensor products ρ1 ⊗ · · · ⊗ ρm where ρi ∈ E, 1 ≤ i ≤ m. Note that if σ1 , σ2 ∈ E1 , then every irreducible subrepresentation of σ1 ⊗ σ2 belongs to E1 . Hence we see that the closed set [E] is exactly the set of all representations in Rep(G) which are equivalent to the sums σ1 ⊕ · · · ⊕ σk where σj ∈ E1 , 1 ≤ j ≤ k.

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