 Polynomial invariant

For a group representation on $V$, a polynomial invariant is a polynomial $p$ satisfying $p(gv)=p(v)$ for every $g\in G$. Homogeneous degree-$d$ invariants form $({\mathrm{Sym}}^d{V}^{\ast }{)}^G$. Using the symmetric power matters: an alternating invariant tensor contracted with repeated copies of the same commuting vector can vanish.