If $A$ is a complex $m$ by $n$ matrix, thus representing a $\mathbb{C}$-linear map from $\mathbb{C}^n$ to $\mathbb{C}^m$, we denote by $s_1, \ldots , s_r$ its singular values. Let $e_k$ denote the $k$-th elementary symmetric polynomial (in its arguments). Is it true that $$ e_k(s_1, \ldots , s_r) = \left\lVert \Lambda^k A \right\rVert_*, $$ where $\Lambda^k A$ is the $k$-th alternating power of $A$, thus representing a $\mathbb{C}$-linear map from $\Lambda^k(\mathbb{C}^n)$ to $\Lambda^k(\mathbb{C}^m)$ (it is thus a complex $\binom{n}{k}$ by $\binom{m}{k}$ matrix), and $\lVert \cdot \rVert_*$ is the nuclear norm (i.e. the sum of the singular values of the corresponding matrix)?
It is just that I have used the left-hand side in one of my articles and now realize it is probably a norm of another (related) matrix. Basically, in that article, I conjectured that what I am calling $A$ here has rank at least $k$ (for some specific $k$). If $A$ happened to have rank less than $k$, then both the left-hand side and right-hand side of the conjectured equality above would vanish. Thus we seem to be looking at some kind of distance between $A$ and the set of matrices of rank less than $k$. Is this intuition justified?
I am not really stuck and I could probably figure these out on my own, but I just thought it would be interesting to share too.
Edit 1: using SVD applied to $A$, one may write: $$ A = U D V^* $$ where $U$, resp. $V$ is an $m$ by $m$, resp. $n$ by $n$, unitary matrix, and $D$ is a diagonal matrix with only nonnegative entries in its diagonal. The positive entries in the diagonal of $D$ are, by definition, the singular values of $A$.
But then we can write $$ \Lambda^k A = \Lambda^k U \, \Lambda^k D \, \Lambda^k V^*. $$ But $\Lambda^k U$, resp. $\Lambda^k V$, is a unitary $\binom{m}{k}$ by $\binom{m}{k}$, resp. $\binom{n}{k}$ by $\binom{n}{k}$, unitary matrix. And $(\Lambda^k V)^* = \Lambda^k(V^*)$. We thus deduce that the singular values of $\Lambda^k A$ are nothing but the products of $k$ of the singular values of $A$ (taking into account multiplicity), for every possible (multi-)set of $k$ singular values of $A$. The conjectured equality thus follows.
There remains the second part of my post. Is there a norm on the space of complex $m$ by $n$ matrices for which $\lVert \Lambda^k A \rVert_*$ is the induced distance between $A$ and the subset of complex $m$ by $n$ matrices having rank less than $k$?
In that spirit, is it true that $\lVert \Lambda^k A \rVert_*$ is the induced distance between $A$ and the subset of complex $m$ by $n$ matrices with rank less than $k$ with respect to the nuclear norm $\lVert . \rVert_*$? This seems non-trivial as a question.