Real Representations

1.1. Constructions. To get from R to C, we can tensor with C. In a more coordinate-free manner, if W is an R-vector space, then its complexification WC := W ⊗R C is a C-vector space. We can view W as an R-subspace of WC by identifying each w ∈ W with w⊗ 1 ∈ WC. Then an R-basis of W is also a C-basis of WC. In particular, WC has the same dimension as W (but is a vector space over a different field). Conversely, we can view C as R if we forget how to multiply by complex scalars that are not real. In a more coordinate-free manner, if V is a C-vector space, then its restriction of scalars is the R-vector space RV with the same underlying abelian group but with only scalar multiplication by real numbers. If v1, . . . , vn is a C-basis of V , then v1, iv1, . . . , vn, ivn is an R-basis of RV . In particular, dim (RV ) = 2 dimV . Also, if V is a C-vector space, then the complex conjugate vector space V has the same underlying group but a new scalar multiplication · defined by λ · v := λ̄v, where λ̄v is defined using the original scalar multiplication. Complexification and restriction of scalars are not inverse constructions. Instead: