Let $V$ be a finite-dimensional vector space over a field $K$. Given a basis $\{v_1,\dotsc,v_n\}$ for $V$, we define the "square span" of the basis to be the subspace of $V\otimes V$ spanned by $v_1\otimes v_1,v_2\otimes v_2,\dotsc,v_n\otimes v_n$.

It seems clear that, for $n\geq 2$, if the square spans of two bases $v=\{v_1,\dotsc,v_n\}$, $v'=\{v_1',\dotsc,v_n'\}$ have non-trivial intersection, then the two bases have proper non-trivial subsets spanning the same space. What happens given three bases $v$, $v'$ and $v''$ such that their square spans, taken together, span a subspace of $V\otimes V$ of dimension less than $3\dim(V)$? What can one say about $v$, $v'$ and $v''$ then?

  • $\begingroup$ If $n=2$ then all pairs of bases have nontrivially intersecting square spans. $\endgroup$ – MTyson Dec 8 '18 at 20:04
  • $\begingroup$ Let's say $n>2$. $\endgroup$ – H A Helfgott Dec 8 '18 at 21:18

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