Constructing New Representations From Old
There are number of ways to combine representations to obtain new representations. Each of these methods involves the application of a construction from linear algebra to representation theory.
- Given two representations ρ1, ρ2 we may construct their direct sum ρ1 ⊕ ρ2 by (ρ1 ⊕ ρ2) (g)(v,w) = (ρ1(g)v, ρ2(g)w).
- The tensor representation of ρ1, ρ2 is defined by (ρ1 ⊗ ρ2) (v ⊗ w) = ρ1(v) ⊗ ρ2(w).
- Let ρ : G → GL(n,C) be a representation. Then ρ induces a representation ρ* on the dual of the vector space Hom(Cn,C). Let f : Cn → C be a linear functional. The representation ρ* is then defined by the rule ρ* (g) (f) = f(ρ(g)−1). The representation ρ* is called either the dual representation or the contragredient representation of ρ.
- Furthermore, if a representation ρ has a subrepresentation σ then the quotient of the representing vector spaces for ρ and σ has a well defined action of G on it. We call the resulting representation the quotient representation of ρ by σ.
Read more about this topic: Representation Theory Of Finite Groups
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