# Prehomogeneous vector spaces

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 $x\in V$.

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Haven't you asked the same question and a very similar one more than a month ago on MO (the second one has even an accepted answere)? Moreover, it wouldn't hurt to recall the definitions, describe your motivation and what exactly the problem is. I can't believe that you haven't made any progress towards answering that question in case you really care about it. –  t.b. May 26 '11 at 10:30
It is true that I asked a similar question, but I was thinking that here there are different users. Just one more comment, I think your last sentence is useless. –  Michele Torielli May 27 '11 at 8:33
It's true there are different users here, but it would be good style to at least link to the other questions and point out what you're still missing, so that people don't start from scratch when a lot of work has already been done. –  joriki May 27 '11 at 9:53