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.