How can I prove that the following 2 prehomogeneous vector spaces are not isomorphic? 1) $(GL_1^2 \times SL_2,2\Lambda_1\oplus\Lambda_1, \mathbb{C}^3\oplus\mathbb{C}^2)$ 2) $(GL_1^2 \times SL_2,2\Lambda_1\oplus\Lambda_1^*, \mathbb{C}^3\oplus\mathbb{C}^2)$? In the case of prehomogeneous vector spaces the notion of isomorphism is given by:
Two triplets $(G, \rho, V)$ and $(G', \rho', V')$ are isomorphic if there exist a rational isomorphism $\sigma : \rho(G) \to \rho'(G')$ and an isomorphism $\tau : V \to V'$, both defined over $\mathbb{C}$, such that $$\tau(\rho(g)x)=\sigma\rho(g)(\tau(x))$$ for all $g\in G$ and $x\in V$.
