An undergraduate student asked me the following seemingly easy question. After a few days of thinking, I still couldn't come up with an answer, nor could I find one online. Maybe folks here could help?
Let $$ S = \{ g \in \mathbf{GL}_n(\mathbb C) \mid \text{ all principal minors of $g$ are nonzero}\},$$ $$ S' = \{ g \in \mathbf{GL}_n(\mathbb C) \mid \exists \text{ permutation matrix } w \text{ such that } gw \in S\}.$$ Of course $S$ and $S'$ are open subvarieties in $\mathbf{GL}_n(\mathbb C)$, and $S' = \bigcup_{w \text{ perm. mat.}} S w$. The question is: does $S$ or $S'$ has any alternative/cleaner/better description? Or, what is complements of $S$ (or $S'$)?
For example, for $n = 3$, $$ g = \begin{pmatrix} 1 & 1 & 1 \\ & 1 & 1\\ 1 & & 1 \end{pmatrix} \in \mathbf{GL}_3(\mathbb C)$$ is not in $S'$.
Thanks in advance!
