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Hello everybody! Define the action of $SL_4({\mathbb{Z}})$ on alternating 2-forms or simply skew-symmetric matrices of degree 4 according to the following: For $B \in SL_4({\mathbb{Z}})$ and an alternating matrix like $M$ define: $B.M = BMB^{T}$ Question: Why the single invariant for this action is $Pf(M)$? Question2: Or why the invariant will be the coefficient of $Pf(Mx-Ny)$ by considering the action of $SL_4({\mathbb{Z}})$ on the pairs of skew-symmetric matrices instead? Thanks ;) |
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One of the coincidences of simple Lie groups is that (over $\mathbb{C}$) $SL(4)=Spin(6)$. From the point of view of $SL(4)$, the three fundamental representations are three exterior powers of the vector representation. From the point of view of $Spin(6)$ they are the vector representation and the two spin representations. You can find the invariant theory of the vector representation of $SO(2n)$ in Weyl's seminal book "Classical groups" and presumably in more recent accounts of this material. I don't know about the passage from $\mathbb{C}$ to $\mathbb{Z}$. |
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