So, here a cute little algebraic question that I came across thinking about a finite groups question Noah asked me (he or I might blog about that later):
Choose your favorite set of m linear polynomials in n variables (if you don’t like coordinates, your favorite m element subset of your favorite n dimensional subspace). When do the the jth powers of these polynomials span all polynomials of degree j?
Of course, for j=1, this is easy. It isn’t immediately clear what happens even for j=2; I think the answer is something like you must contain a subset of polynomials such that no k dimensional subspace contains more than of them. Any thoughts? (Also, special bonus bonus points to whoever figures out what representation theory question this is supposed to solve).