Cartesian Sums of Representations

For example, consider the representations (5U (2), P , Rn) for n = 0, 1, 2. The vector space P P P is the set of complex-coefficient polynomials in two variables of degree two or less. Specifically, the polynomial co+cio- c + Ciiy + C iQx + C2ixy + C22y2 corresponds to the element 

Recall the projection onto the k-th summand from Definition 2.12. This projection is a homomorphism of representations. 

Proposition 5.6 Suppose (G, Vk, Pk) representations for each k = Then the projection 

Physically, expressing a vector space of wave functions as a Cartesian sum 

---