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The title says it all. For the uninitiated: Any map $f:V \to W$ induces a map $\wedge^k V \to \wedge^k W$ by $v_1 \wedge \cdots \wedge v_k \mapsto f(v_1)\wedge \cdots \wedge f(v_k)$, so $\wedge^k(-)$ is a functor from vector spaces to itself.

I have a map $\varphi:\wedge^k V\to \wedge^kV$ and I have reason to suspect that this map does not come fram a map $\psi:V \to V$.

I'm not sure how to prove this. It seems that any map in the image of $\wedge^k(-)$ must satisfy some kind of Plücker relations (similar to those of the Grassmannian), since if $\varphi \in \hom(V,V)$ is represented by the matrix $(x_{ij})$, then its image in $\hom(\wedge^k V,\wedge^k V)$ has matrix $(\det_{IJ}(x_{ij}))$, where $\det_{IJ}$ means take the determinant of the submatrix with indices $I$ and $J$.

I tried asking Macaulay2 to compute the ideal of relations, but even in the case $\dim V=4$ and $k=2$, it does not seem to be a feasible computation:

R = QQ[x_1..x_16]
M = genericMatrix(R,4,4)
I = minors(2,M)
numgens I
>> 36
S = QQ[y_1..y_36]
f = map(R,S,gens I)
ker f
>> ......???? <- to much for M2!

Is there some strategy to determine if my map comes from $\hom(V,V)$ other than computing these relations? If not, are the relations between the $k\times k$-minors of a $n\times n$-matrix known?

Thanks.

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Interesting question. In your matrix description $I,J$ have cardinality $k$, right? And there is no harm in replacing $\hom(V,V)$ by $\hom(V,W)$, right? –  Martin Brandenburg Mar 15 '13 at 14:37
    
@Martin: Yes, you're right. $I,J$ runs through all subsets cardinality $k$. –  Fredrik Meyer Mar 15 '13 at 14:39
    
If $V$ has dimension $d$ and $k=d-1$, then there is a canonical(!) isomorphism $\wedge^{d-1} V \cong V^* \otimes \wedge^d V$, and $\wedge^d V$ is invertible. In particular we get a map $\wedge^{d-1} : \mathrm{End}(V) \to \mathrm{End}(V^*)$. It identifies with $f \mapsto f^*$ and is therefore an isomorphism. So interesting things only happen for $1<k<d-1$. –  Martin Brandenburg Mar 15 '13 at 14:48
    
if $k \le \frac{1}{2}\dim V$ then you can try computing things like $\phi(v_1 \wedge v_{i_1} \wedge \cdots \wedge v_{i_{k-1}}) \wedge \phi(v_1 \wedge v_{j_1} \wedge \cdots \wedge v_{j_{k-1}})$. These have to be zero if $\phi$ comes from a map $V \to V$. –  Eric O. Korman Mar 18 '13 at 12:59
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1 Answer

up vote 2 down vote accepted

The (Zariski closure of the) image of your map is studied in the following paper of Bruns and Conca:

http://arxiv.org/abs/0705.3399

They stratify this closure by two numerical invariants rank and "small rank" and show that they correspond to $GL(V) \times GL(W)$ orbits. I never read the paper carefully, so I don't know if it will be of any use, but hopefully there are some interesting things in there for you.

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