The elementary tensors $x\otimes \cdots \otimes x$, with all factors equal, are all in the subspace $\vee^k\scrH$. Do they span it?

Solution. Yes. Indeed, take $$\beex \bea &\quad (x+y)\otimes (x+y)-x\otimes x-y\otimes y\\ &=x\otimes y+y\otimes x\\ &=\sqrt{2!}x\vee y \eea \eeex$$ for example.

[Bhatia.Matrix Analysis.Solutions to Exercises and Problems]ExI.5.2