Automorphisms of the ring of invariants of the binary quintic representation of SL2

Abstract

Let k^[6] denote a polynomial ring in 6 variables over an algebraically closed field k of characteristic zero and consider the action of SL2(k) on k^[6] induced by the irreducible representation of SL2 of degree 5 (the binary quintic representation). We consider the ring Q = (k^[6])^SL2 of invariant polynomials and show that Aut_k(Q) = u(k), the unit group of k, where Aut_k(Q) is the group of k-algebra automorphisms of Q. Based on this result, we show that the group of SL2-equivariant polynomial automorphisms of k^[6] is isomorphic to u(k).