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The result is true if r=2 , see Le Potier, J. Stabilité et amplitude sur P2. in Progress in Math., 7 (1980), 145–182, Birkhauser. see also Anghel, C., Coanda, I., Manolache, N. Globally Generated Vector Bundles on P^n with c_1=4 Ellia, P. Chern classes of rank two globally generated vector bundles on P2. Rend. Lincei Mat. Appl. 24 (2013), 147–163

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Ad 1.: Since every vector can be decomposed in its horizontal and vertical part. Thus it is enough to consider the case a) where all vectors are horizontal (this is trivial) and b) where at least one is vertical and the rest is horizontal (this has to be calculated explicitly but is relatively simple since one can use the fact that the vertical vector is a ...

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These questions are all answered in terms of the exterior algebra over the ring $R$ of the free module $M=R^n$. Namely, the skew-symmetric matrices naturally live in $\Lambda^2(M) \simeq R^N$ where $N=\tfrac12n(n{-}1)$. The top exterior power is $\Lambda^n(M)\simeq R$. If $n=2m$, and $A\in\Lambda^2(M)$ is given, then one finds that \$A^m = m!\, ...

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