Consider a Grassmannian $G=Gr(r,n)$ embedded in projective space $P^n$ by its Plucker embedding. Is there a way of writing down a presentation of the tautological bundle of $G$, as a module over the coordinate ring?
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1$\begingroup$ Which tautological bundle: the quotient bundle of the trivial bundle or the subbundle of the trivial bundle? $\endgroup$– Jason StarrMar 14, 2015 at 18:41
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$\begingroup$ Primarily the subbundle, but I'm interested doing it for both. $\endgroup$– gsvrMar 16, 2015 at 6:51
1 Answer
The Grassmanian is the quotient of $k\times n$ matrices by the left action of $GL_k$. You want to take a $k$-dimensional vector space and quotient by the diagonal action of $GL_k$ on both. So you can take a free module of rank $k$ over the free ring on $kn$ variables and quotient by the action of $GL_k$. You can get generators by viewing it as a $k \times n+1$ matrix and taking minors that include the last column. The relations will be the relations on Plucker coordinates of $Gr(k,n+1)$. This gives a sub bundle of the trivial bundle.
For the quotuent bundle, there are $n$ obvious generators. There is a relation among any $n-k+1$ of them. To get the relation it's better to view the matrix as $n-k \times n$. Take the selected columns and add a row of free variables, making it square. The relation expresses that the determinant vanishes using the formula for determinant involving one row and the minors