How can I prove that the following 2 prehomogeneous vector spaces are not isomorphic? 1)$(GL_n,\Lambda_1\oplus \Lambda_1,\mathbb{C}^n \oplus \mathbb{C}^n)$ 2)$(GL_n,\Lambda_1\oplus \Lambda_1^*,\mathbb{C}^n \oplus \mathbb{C}^n)$ where $\Lambda_1$ is the standard representation of $GL_n$ on $\mathbb{C}^n$. 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 $x\in V$. That is the following diagram is commutative for all $g\in G$:

$$\require{AMScd} \begin{CD} V @>{\tau}>> V'\\ @V{\rho(g)}VV @VV{\sigma\rho(g)}V \\ V @>{\tau}>> V' \end{CD} $$