Let \(k^{[6]}\) denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of \({{\,\textrm{SL}\,}}_2(k)\) on \(k^{[6]}\) induced by the irreducible representation of \({{\,\textrm{SL}\,}}_2\) of degree 5 (the binary quintic representation). We consider the ring \(Q = (k^{[6]})^{{{\,\textrm{SL}\,}}_2}\) of invariant polynomials and show that \(\textrm{Aut}_k(Q) = k^*\) , where \(\textrm{Aut}_k(Q)\) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of \({{\,\textrm{SL}\,}}_2\) -equivariant polynomial automorphisms of \(k^{[6]}\) is isomorphic to \(k^*\) .
