Suppose $f$ is a linear map, and consider $\Lambda^k f$ as the usual exterior power of $f$ (if you prefer matrices, it is a matrix whose entries are the $k\times k$ minors of $f.$) The coefficients of the characteristic polynomial of $f$ are the traces of the various $\Lambda^k f,$ but my question is: is there a simple (for your favorite value of "simple") way to write the characteristic polynomial of $\Lambda^k f$ in terms of $f$ (ideally, in terms of the characteristic polynomial of $f,$ but I am not certain that is possible).



Characteristic polynomial of exterior power
I am not sure what data about $f$ you could use other than the characteristic polynomial. The $l$th coefficient of the characteristic polynomial of $\Lambda^k f$ is the trace of $\Lambda^l (\Lambda^k f)$. You would like to write the trace of $\Lambda^l (\Lambda^k f)$ in terms of the trace of $\Lambda^i f$, $1\leq i \leq n$. The unique way to do this is equivalent to describing $\Lambda^l (\Lambda^k V)$ in the representation ring of $GL(V)$ as a polynomial in the $\Lambda^i(V)$, which generate that ring.


So the formulas you are looking for are exactly the polynomials $P\_{m,n}$ in the definition of a $\lambda$-ring: <http://en.wikipedia.org/wiki/%CE%9B-ring>


There they are expressed in terms of symmetric functions in a way that is pretty simple to state but pretty complicated to do computations with. I don't know any clever combinatorial way to simplify these computations, and it seems like a hard combinatorics problem in general. (If you just wanted to get to the symmetric polynomial picture, then going through representation theory might be pointless, as you could just assume $f$ was diagonalizable and go from there.)


Alternately, you could choose to write the coefficients in terms of the eigenvalues of $f$, and then you would write them as a sum over subsets of size $l$ of the set of subsets of size $k$ of the eigenvalues of $f$. You might find this description more elegant.


Finally you could express the $\lambda$-operations in terms of [Adams operations](http://en.wikipedia.org/wiki/Adams_operation) by writing the elementary symmetric polynomials in terms of Newton symmetric polynomials. I think this is the easiest way to do computations and remember them and small $k$ and $l$, and is probably the most mathematically interesting way because it is connected to [number theory](http://arxiv.org/abs/0906.3146).


