Characterising minors of diagonal matrices Let $k,d$ be positive integers, $1<k<d$. Let $\lambda_I=\lambda_{i_1,\ldots,i_k}$ be real numbers, indexed by multi-indices $I=(i_1,\ldots,i_k)$, where $1\le i_1<\ldots<i_k \le d$.
Are there necessary and sufficient conditions on $\lambda_{i_1,\ldots,i_k}$ which are equivalent to the existence of $\sigma_1,\ldots,\sigma_d \in \mathbb{R}$ such that $\lambda_{i_1,\ldots,i_k}=\sigma_{i_1}\cdot \ldots\cdot\sigma_{i_k}$ holds for every multi-index $I$?

In other words, I am asking whether we can characterise which sequences of real numbers can arise as the $k$-minors of diagonal $d \times d$ matrices?

I am interested mainly in the case where all the $\lambda_{i_1,\ldots,i_k}$ are non-zero.
I have heard that the general problem of recognizing $k$-minors of arbitrary square matrices is open, but I am hoping that for diagonal matrices, the situation maybe better understood.
I guess this should be easier by starting over $\mathbb{C}$. What is known about that case?
Commnet: If I understand correctly, the Plucker relations only describe the minors of top-degree of a non-square matrix. Here I am talking about the minors of degree $k$, when $1<k<d$, i.e. non-top minors of a square matrix.
 A: Let me give an answer over complex numbers.
The conditions are the following: for each pair $i \ne j$ and for each pair of subsets $I,J \subset \{1,\dots,d\} \setminus \{i,j\}$ of cardinality $k - 1$ one has a relation
$$
\lambda_{I \sqcup \{i\}} \cdot \lambda_{J \sqcup \{j\}} = 
\lambda_{I \sqcup \{j\}} \cdot \lambda_{J \sqcup \{i\}}.
$$
Clearly these relations are necessary.
Let us also show that they are sufficient. In fact, let me just explain how $\sigma_i$ can be reconstructed. Take any set $J \subset \{1,\dots,d-1\}$ of cardinality $k$. Then set
$$
\sigma_d = \sqrt[k]{\frac{\prod_{j \in J} \lambda_{J \setminus \{j\} \sqcup \{d\}}}{\lambda_J^{k-1}}}.
$$
After that for each $i \in \{1,\dots,d-1\}$ choose $J \subset \{1,\dots,d-1\} \setminus \{i\}$ of cardinality $k-1$ and set
$$
\sigma_i = \frac{\prod_{j \in J} \lambda_{J \setminus \{j\} \sqcup \{i,d\}}}{\lambda_{J \sqcup \{i\}}^{k-2}\sigma_d^{k-1}}.
$$
One can check that this solves the required equations.
Over real numbers the only extra problem is the existence of the root.
